The NP-complete problems Colouring and k-Colouring (k ≥ 3) are well studied on H-free graphs, i.e., graphs that do not contain some fixed graph H as an induced subgraph. We research to what extent the known polynomial-time algorithms for H-free graphs can be generalized if we only know some of the edges of the input graph. We do this by considering the classical probe graph model introduced in the early nineties. For a graph H, a partitioned probe H-free graph (G,P,N) consists of a graph G = (V,E), together with a set P ⊆ V of probes and an independent set N = V ⧵ P of non-probes, such that G+F is H-free for some edge set F ⊆ binom(N,2). We show the following: - We fully classify Colouring on partitioned probe H-free graphs and show that the obtained complexity dichotomy differs from the known dichotomy of Colouring for H-free graphs. - We fully classify 3-Colouring on partitioned probe P_t-free graphs: we prove polynomial-time solvability for t ≤ 5 and NP-completeness for t ≥ 6. In contrast, 3-Colouring on P_t-free graphs is known to be polynomial-time solvable for t ≤ 7 and quasi-polynomial-time solvable for t ≥ 8. Our main result is our polynomial-time algorithm for 3-Colouring on partitioned P₅-free graphs. For this result, and also for all our other polynomial-time results, we do not need to know the edge set F; we only need to know its existence. Moreover, the class of probe P₅-free graphs includes not only paths of arbitrary length but even all bipartite graphs and is much richer than the class of P₅-free graphs. The latter is also evidenced by the fact that there exist graph problems, such as Matching Cut, that are known to be polynomial-time solvable for P₅-free graphs but NP-complete for partitioned probe P₅-free graphs. In particular, unlike the class of 3-colourable P₅-free graphs, the class of 3-colourable probe P₅-free graphs has unbounded mim-width. Hence, our polynomial-time result for 3-Colouring for probe P₅-free graphs suggests that there may be another, deeper overarching reason why 3-Colouring is polynomial-time solvable for P₅-free graphs.

In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number, which fit into a framework based on whether every colour class has (i) at least one b-chromatic vertex, (ii) exactly one b-chromatic vertex, or (iii) all of its vertices being b-chromatic. By combining known and new results, we fully classify the computational complexity of b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number in H-free graphs. For Tight b-Chromatic Number in H-free graphs, we develop a general technique to determine new graphs H, for which the problem is polynomial-time solvable, and we also determine new graphs H, for which the problem is still NP-complete. We show, for the first time, the existence of a graph H such that in H-free graphs, b-Chromatic Number is NP-hard, while Tight b-Chromatic Number is polynomial-time solvable.

A graph class is monotone if it is closed under taking subgraphs. A monotone class defined by finitely many obstructions has bounded treewidth if and only if one of the obstructions is a tripod, i.e. a disjoint union of subdivided claws and paths. This dichotomy also characterizes exactly those monotone graph classes for which many NP-hard graph problems admit polynomial-time algorithms. These dichotomies do not extend to the universe of all hereditary classes. This leads to the question of whether we can extend known dichotomies for monotone classes to larger families of hereditary classes. We answer this question affirmatively by considering the family of hereditary graph classes closed under self-intersection. This family is known to be located strictly between the monotone and hereditary classes. We prove a new structural characterization of graphs in self-intersection-closed classes excluding a tripod. In contrast to monotone classes excluding a tripod, these classes do not necessarily have bounded treewidth; in fact, they do not even need to be sparse. We use our characterization to give a complete dichotomy for Maximum Independent Set, and its weighted variant, on self-intersection-closed classes defined by finitely many obstructions: these problems are in P if the class excludes a tripod and NP-hard otherwise. Our dichotomy generalizes several known results on Maximum Independent Set in the literature. We also apply our characterization to obtain a dichotomy for Maximum Induced Matching on self-intersection-closed classes of bipartite graphs defined by finitely many obstructions, and for Satisfiability and Counting Satisfiability on self-intersection-closed classes of (bipartite) incidence graphs defined by finitely many obstructions. Finally, we use our characterization to obtain a dichotomy for boundedness of clique-width for self-intersection-closed classes of bipartite graphs defined by finitely many obstructions.

We consider Colouring on graphs that are $H$-subgraph-free for some fixed graph $H$, which are graphs that do not contain $H$ as a subgraph. To classify the complexity of Colouring on $H$-subgraph-free graphs for connected $H$, it remains to consider when $H$ is a tree of maximum degree $4$ with exactly one vertex of degree $4$, or a tree of maximum degree $3$ with at least two vertices of degree $3$. We let $H$ be a so-called subdivided ``H''-graph, which is either a subdivided $\mathbb{H}_0$: a tree of maximum degree $4$ that is a star, or a subdivided $\mathbb{H}_1$: a tree of maximum degree $3$ with exactly two vertices of degree $3$. We develop new decomposition theorems resulting in polynomial-time algorithms, and in combination with known results, fully classify all cases $\mathbb{H}_0$ and $\mathbb{H}_1$. To illustrate the wider applicability of our techniques, we also employ them to obtain similar new polynomial-time results for two other classic graph problems: Stable Cut and, in part, Feedback Vertex Set.

The Matching Cut problem is to decide if the vertex set of a connected graph can be partitioned into two non-empty sets B and R such that the edges between B and R form a matching, that is, every vertex in B has at most one neighbour in R, and vice versa. If for some integer d ≥ 1, we allow every vertex in B to have at most d neighbours in R, and vice versa, we obtain the more general problem d-Cut. It is known that d-Cut is NP-complete for every d ≥ 1. However, for claw-free graphs, it is only known that d-Cut is polynomial-time solvable for d = 1 and NP-complete for d ≥ 3. We resolve the missing case d = 2 by proving NP-completeness. This follows from our more general study, in which we also bound the maximum degree. That is, we prove that for every d ≥ 2, d-Cut, restricted to claw-free graphs of maximum degree p, is constant-time solvable if p ≤ 2d+1 and NP-complete if p ≥ 2d+3. Moreover, in the former case, we can find a d-cut in linear time. We also show how our positive results for claw-free graphs can be generalized to S_{1^t,𝓁}-free graphs where S_{1^t,𝓁} is the graph obtained from a star on t+2 vertices by subdividing one of its edges exactly 𝓁 times.

Kidney Exchange Programs (KEPs) facilitate the exchange of kidneys, and larger pools of recipient-donor pairs tend to yield proportionally more transplants, leading to the proposal of international KEPs (IKEPs). However, as studied by Mincu et al. [2021], practical limitations must be considered in IKEPs to ensure that countries remain willing to participate. Thus, we study IKEPs with country-specific parameters, represented by a tuple Γ, restricting the selected transplants to be feasible for the countries to conduct, e.g., imposing an upper limit on the number of consecutive exchanges within a country's borders. We provide a complete complexity dichotomy for the problem of finding a feasible (according to the constraints given by Γ) cycle packing with the maximum number of transplants, for every possible Γ. We also study the potential for countries to misreport their parameters to increase their allocation. As manipulation can harm the total number of transplants, we propose a novel individually rational and incentive compatible mechanism Morder. We first give a theoretical approximation ratio for Morder in terms of the number of transplants, and show that the approximation ratio of Morder is asymptotically optimal. We then use simulations which suggest that, in practice, the performance of Morder is significantly better than this worst-case ratio.

Abstract The (Perfect) Matching Cut problem is to decide if a connected graph has a (perfect) matching that is also an edge cut. The Disconnected Perfect Matching problem is to decide if a connected graph has a perfect matching that contains a matching cut. Both Matching Cut and Disconnected Perfect Matching are -complete for planar graphs of girth 5, whereas Perfect Matching Cut is known to be -complete even for subcubic bipartite graphs of arbitrarily large fixed girth. We prove that Matching Cut and Disconnected Perfect Matching are also -complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Our result for Matching Cut resolves a 20-year old open problem. We also show that the more general problem d -Cut, for every fixed $$d\ge 1$$ d ≥ 1 , is -complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Furthermore, we show that Matching Cut, Perfect Matching Cut and Disconnected Perfect Matching are -complete for H-free graphs whenever H contains a connected component with two vertices of degree at least 3. Afterwards, we update the state-of-the-art summaries for H-free graphs and compare them with each other, and with a known and full classification of the Maximum Matching Cut problem, which is to determine a largest matching cut of a graph G. Finally, by combining existing results, we obtain a complete complexity classification of Perfect Matching Cut for $$\mathcal{H}$$ H -subgraph-free graphs where $$\mathcal{H}$$ H is any finite set of graphs.

Abstract For a set of graphs $${\mathcal {H}}$$ H , a graph G is $${\mathcal {H}}$$ H -subgraph-free if G does not contain any graph from $${{{\mathcal {H}}}}$$ H as a subgraph. We propose general and easy-to-state conditions on graph problems that explain a large set of results for $${\mathcal {H}}$$ H -subgraph-free graphs. Namely, a graph problem must be efficiently solvable on graphs of bounded treewidth, computationally hard on subcubic graphs, and computational hardness must be preserved under edge subdivision of subcubic graphs. Our meta-classification says that if a graph problem $$\Pi $$ Π satisfies all three conditions, then for every finite set $${{{\mathcal {H}}}}$$ H , it is “efficiently solvable” on $${{{\mathcal {H}}}}$$ H -subgraph-free graphs if $${\mathcal {H}}$$ H contains a disjoint union of one or more paths and subdivided claws, and $$\Pi $$ Π is “computationally hard” otherwise. We apply our meta-classification on many well-known partitioning, covering and packing problems, network design problems and width parameter problems to obtain a dichotomy between polynomial-time solvability and -completeness. For distance-metric problems, we obtain a dichotomy between almost-linear-time solvability and having no subquadratic-time algorithm (conditioned on some hardness hypotheses). Apart from capturing a large number of explicitly and implicitly known results in the literature, we also prove a number of new results. Moreover, we perform an extensive comparison between the subgraph framework and the existing frameworks for the minor and topological minor relations, and pose several new open problems and research directions.

We study Steiner Forest on H -subgraph-free graphs, that is, graphs that do not contain some fixed graph H as a (not necessarily induced) subgraph. In contrast to the related Steiner Tree problem, Steiner Forest falls outside a recent framework that completely characterizes the complexity of many problems on H -subgraph-free graphs. Hence, the complexity of Steiner Forest on H -subgraph-free graphs remained open. Our main results are four polynomial-time algorithms for different excluded graphs H that are central to further understand its complexity. We also study the complexity of Steiner Forest for graphs with a small c -deletion set, that is, a small set X of vertices such that each connected component of G − X has size at most c . For this parameter, we give two algorithms that we later employ as subroutines (including a faster algorithm when c = 1 , that is, the vertex cover number) and exhibit a dichotomy theorem.

A (proper) colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. The corresponding decision problems are Acyclic Colouring , Star Colouring and Injective Colouring . We give almost complete complexity classifications for Acyclic Colouring , Star Colouring and Injective Colouring on H -free graphs (for each of the problems, we have one open case). Moreover, we give full complexity classifications if the number of colours k is fixed, that is, not part of the input. From our study it follows that for fixed k , the three problems behave in the same way, but this is no longer true if k is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs of multigraphs.

Abstract. The intensively studied Diameter problem is to find the diameter of a given connected graph. We investigate, for the first time in a structured manner, the complexity of Diameter for [Formula: see text]-free graphs, that is, graphs that do not contain a fixed graph [Formula: see text] as an induced subgraph. We first show that if [Formula: see text] is not a linear forest with small components, then Diameter cannot be solved in subquadratic time for [Formula: see text]-free graphs under SETH. For some small linear forests, we do show linear-time algorithms for solving Diameter. For other linear forests [Formula: see text], we make progress towards linear-time algorithms by considering specific diameter values. If [Formula: see text] is a linear forest, the maximum value of the diameter of any graph in a connected [Formula: see text]-free graph class is some constant [Formula: see text] dependent only on [Formula: see text]. We give linear-time algorithms for deciding if a connected [Formula: see text]-free graph has diameter [Formula: see text] for several linear forests [Formula: see text]. In contrast, for one such linear forest [Formula: see text], Diameter cannot be solved in subquadratic time for [Formula: see text]-free graphs under SETH. Moreover, we even show that, for several other linear forests [Formula: see text], one cannot decide in subquadratic time if a connected [Formula: see text]-free graph has diameter [Formula: see text] under SETH.

We consider a natural generalization of Vertex Cover : the Subset Vertex Cover problem, which is to decide for a graph G = ( V , E ) , a subset T ⊆ V and integer k , if V has a subset S of size at most k , such that S contains at least one end-vertex of every edge incident to a vertex of T . A graph is H -free if it does not contain H as an induced subgraph. We solve two open problems from the literature by proving that Subset Vertex Cover is NP -complete on subcubic (claw, diamond)-free planar graphs and on 2-unipolar graphs, a subclass of 2 P 3 -free weakly chordal graphs. Our results show for the first time that Subset Vertex Cover is computationally harder than Vertex Cover (under P ≠ NP ). We also prove new polynomial time results, some of which follow from a reduction to Vertex Cover restricted to classes of probe graphs. We first give a dichotomy on graphs where G [ T ] is H -free. Namely, we show that Subset Vertex Cover is polynomial-time solvable on graphs G , for which G [ T ] is H -free, if H = s P 1 + t P 2 and NP -complete otherwise. Moreover, we prove that Subset Vertex Cover is polynomial-time solvable for ( s P 1 + P 2 + P 3 ) -free graphs and bounded mim-width graphs. By combining our new results with known results we obtain a partial complexity classification for Subset Vertex Cover on H -free graphs.

Abstract To overcome incompatibility issues, kidney patients may swap their donors. In international kidney exchange programmes (IKEPs), countries merge their national patient–donor pools. We consider a recently introduced credit system. In each round, countries are given an initial “fair” allocation of the total number of kidney transplants. This allocation is adjusted by a credit function yielding a target allocation. The goal is to find a solution that approaches the target allocation as closely as possible, to ensure long-term stability of the international pool. As solutions, we use maximum matchings that lexicographically minimize the country deviations from the target allocation. We perform, for the first time, a computational study for a large number of countries. For the initial allocations we use two easy-to-compute solution concepts, the benefit value and the contribution value, and four classical but hard-to-compute concepts, the Shapley value, nucleolus, Banzhaf value and tau value. By using state-of-the-art software we show that the latter four concepts are now within reach for IKEPs of up to fifteen countries. Our experiments show that using lexicographically minimal maximum matchings instead of ones that only minimize the largest deviation from the target allocation (as previously done) may make an IKEP up to 54% more balanced.

A bipartite graph G=(A,B,E) is H-convex for some family of graphs H if there exists a graph H∈H with V(H)=A such that the neighbours in A of each b∈B induce a connected subgraph of H. Many NP-complete problems are polynomial-time solvable for H-convex graphs when H is the set of paths. The underlying reason is that the class has bounded mim-width. We extend this result to families of H-convex graphs where H is the set of cycles, or H is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least 3. As a consequence, we strengthen many known results via one general and short proof. We also show that the mim-width of H-convex graphs is unbounded if H is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least 3.

Abstract. A homomorphism [Formula: see text] from a guest graph [Formula: see text] to a host graph [Formula: see text] is locally bijective, injective, or surjective if for every [Formula: see text], the restriction of [Formula: see text] to the neighbourhood of [Formula: see text] is bijective, injective, or surjective, respectively. We prove a number of new FPT (fixed-parameter tractable), W [1]-hard, and paraNP -complete results for the corresponding decision problems LBHom, LIHom, and LSHom by considering a hierarchy of parameters of the guest graph [Formula: see text]. In this way we strengthen several existing results. For our FPT results, we develop a new algorithmic framework that involves a general ILP (integer linear program) model. We also use our framework to prove FPT results for the Role Assignment problem, which originates from social network theory and is closely related to locally surjective homomorphisms.

Matching cut Perfect matching 𝐻-free graph Diameter Radius DichotomyThe (Perfect) Matching Cut problem is to decide if a graph 𝐺 has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of 𝐺.Both Matching Cut and Perfect Matching Cut are known to be NP-complete.A perfect matching cut is also a matching cut with maximum number of edges.To increase our understanding of the relationship between the two problems, we perform a complexity study for the Maximum Matching Cut problem, which is to determine a largest matching cut in a graph.Our results yield full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and 𝐻-free graphs.A disconnected perfect matching of a graph 𝐺 is a perfect matching that contains a matching cut of 𝐺.We also show how our new techniques can be used for finding a disconnected perfect matching with a largest matching cut for special graph classes.In this way we can prove that the decision problem Disconnected Perfect Matching is polynomial-time solvable for (𝑃 6 + 𝑠𝑃 2 )-free graphs for every 𝑠 ≥ 0, extending a known result for 𝑃 5 -free graphs (Bouquet and Picouleau, 2020).

International audience

For any finite set H = {H1, . . ., Hp} of graphs, a graph is H-subgraph-free if it does not contain any of H1, . . ., Hp as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity can be classified on classes of H-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most 3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set.

The (Perfect) Matching Cut problem is to decide if a graph $G$ has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of $G$. Both Matching Cut and Perfect Matching Cut are known to be NP-complete. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we perform a complexity study for the Maximum Matching Cut problem, which is to determine a largest matching cut in a graph. Our results yield full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and $H$-free graphs. A disconnected perfect matching of a graph $G$ is a perfect matching that contains a matching cut of $G$. We also show how our new techniques can be used for finding a disconnected perfect matching with a largest matching cut for special graph classes. In this way we can prove that the decision problem Disconnected Perfect Matching is polynomial-time solvable for $(P_6+sP_2)$-free graphs for every $s\geq 0$, extending a known result for $P_5$-free graphs (Bouquet and Picouleau, 2020).

Abstract The well-known -complete problem Matching Cut is to decide if a graph has a matching that is also an edge cut of the graph. We prove new complexity results for Matching Cut restricted to H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. We also prove new complexity results for two recently studied variants of Matching Cut, on H-free graphs. The first variant requires that the matching cut must be extendable to a perfect matching of the graph. The second variant requires the matching cut to be a perfect matching. In particular, we prove that there exists a small constant $$r>0$$ r > 0 such that the first variant is -complete for $$P_r$$ P r -free graphs. This addresses a question of Bouquet and Picouleau (The complexity of the Perfect Matching-Cut problem. CoRR, arXiv:2011.03318 , (2020)). For all three problems, we give state-of-the-art summaries of their computational complexity for H-free graphs.

Abstract Paths $$P^1,\ldots ,P^k$$ P 1 , … , P k in a graph $$G=(V,E)$$ G = ( V , E ) are mutually induced if any two distinct $$P^i$$ P i and $$P^j$$ P j have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices $$(s_i,t_i)$$ ( s i , t i ) contains k mutually induced paths $$P^i$$ P i such that each $$P^i$$ P i starts from $$s_i$$ s i and ends at $$t_i$$ t i . This is a classical graph problem that is -complete even for $$k=2$$ k = 2 . We introduce a natural generalization, Induced Disjoint Connected Subgraphs: instead of connecting pairs of terminals, we must connect sets of terminals. We give almost-complete dichotomies of the computational complexity of both problems for H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. Finally, we give a complete classification of the complexity of the second problem if the number k of terminal sets is fixed, that is, not part of the input.

Matching games naturally generalize assignment games, a well-known class of cooperative games. Interest in matching games has grown recently due to some breakthrough results and new applications. This state-of-the-art survey provides an overview of matching games and extensions, such as b-matching games and partitioned matching games; the latter originating from the emerging area of international kidney exchange. In this survey we focus on computational complexity aspects of various game-theoretical solution concepts, such as the core, nucleolus and Shapley value, when the input is restricted to a matching game or one of its variants.

Paths P1,…,Pk in a graph G=(V,E) are mutually induced if any two distinct Pi and Pj have neither common vertices nor adjacent vertices. For a fixed integer k, the k-Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that each Pi starts from si and ends at ti. Whereas the non-induced version is well-known to be polynomial-time solvable for every fixed integer k, a classical result from the literature states that even 2-Induced Disjoint Paths is NP-complete. We prove new complexity results for k-Induced Disjoint Paths if the input is restricted to H-free graphs, that is, graphs without a fixed graph H as an induced subgraph. We compare our results with a complexity dichotomy for Induced Disjoint Paths, the variant where k is part of the input.

Perfect Matching-Cut is the problem of deciding whether a graph has a perfect matching that contains an edge-cut. We show that this problem is NP-complete for planar graphs with maximum degree four, for planar graphs with girth five, for bipartite five-regular graphs, for graphs of diameter three and for bipartite graphs of diameter four. We show that there exist polynomial time algorithms for the following classes of graphs: claw-free, $P_5$-free, diameter two, bipartite with diameter three and graphs with bounded tree-width.

We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on $H$-free graphs, that is, graphs that do not contain some fixed graph $H$ as an induced subgraph. In particular, we prove that for every $s\geq 1$, both problems are polynomial-time solvable for $sP_3$-free graphs and $(sP_1+P_5)$-free graphs; here, the graph $sP_3$ denotes the disjoint union of $s$ paths on three vertices and the graph $sP_1+P_5$ denotes the disjoint union of $s$ isolated vertices and a path on five vertices. Our new results for Feedback Vertex Set extend all known polynomial-time results for Feedback Vertex Set on $H$-free graphs, namely for $sP_2$-free graphs [Chiarelli et al., Theoret. Comput. Sci., 705 (2018), pp. 75--83], $(sP_1+P_3)$-free graphs [Dabrowski et al., Algorithmica, 82 (2020), pp. 2841--2866] and $P_5$-free graphs [Abrishami et al., Induced subgraphs of bounded treewidth and the container method, in Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadelphia, 2021, pp. 1948--1964]. Together, the new results also show that both problems exhibit the same behavior on $H$-free graphs (subject to some open cases). This is in part due to a new general algorithm we design for finding in a ($sP_3)$-free or $(sP_1+P_5)$-free graph $G$ a largest induced subgraph whose blocks belong to some finite class ${\cal C}$ of graphs. We also compare our results with the state-of-the-art results for the Odd Cycle Transversal problem, which is known to behave differently on $H$-free graphs.

For a connected graph G=(V,E), a matching M⊆E is a matching cut of G if G−M is disconnected. It is known that for an integer d, the corresponding decision problem Matching Cut is polynomial-time solvable for graphs of diameter at most d if d≤2 and NP-complete if d≥3. We prove the same dichotomy for graphs of bounded radius. For a graph H, a graph is H-free if it does not contain H as an induced subgraph. As a consequence of our result, we can solve Matching Cut in polynomial time for P6-free graphs, extending a recent result of Feghali for P5-free graphs. We then extend our result to hold even for (sP3+P6)-free graphs for every s≥0 and initiate a complexity classification of Matching Cut for H-free graphs.

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem \( \mathrm{QCSP}(\mathrm{H}) \) when \( \mathrm{H} \) is a reflexive tournament. It is well known that reflexive tournaments can be split into a sequence of strongly connected components \( \mathrm{H}_1,\ldots ,\mathrm{H}_n \) so that there exists an edge from every vertex of \( \mathrm{H}_i \) to every vertex of \( \mathrm{H}_j \) if and only if \( i\lt j \) . We prove that if \( \mathrm{H} \) has both its initial and final strongly connected component (possibly equal) of size 1, then \( \mathrm{QCSP}(\mathrm{H}) \) is in \( \mathsf {NL} \) and otherwise \( \mathrm{QCSP}(\mathrm{H}) \) is \( \mathsf {NP} \) -hard.

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well-known that reflexive tournaments can be split into a sequence of strongly connected components H₁,…,H_n so that there exists an edge from every vertex of H_i to every vertex of H_j if and only if i < j. We prove that if H has both its initial and final strongly connected component (possibly equal) of size 1, then QCSP(H) is in NL and otherwise QCSP(H) is NP-hard.

A (proper) colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as $L(1,1)$-Labelling). A classical complexity result on Colouring is a well-known dichotomy for $H$-free graphs (a graph is $H$-free if it does not contain $H$ as an induced subgraph). In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years. We perform such a study and give almost complete complexity classifications for Acyclic Colouring, Star Colouring and Injective Colouring on $H$-free graphs (for each of the problems, we have one open case). Moreover, we give full complexity classifications if the number of colours $k$ is fixed, that is, not part of the input. From our study it follows that for fixed $k$ the three problems behave in the same way, but this is no longer true if $k$ is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs of multigraphs.

We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. In particular, we prove that both problems are polynomial-time solvable for sP₃-free graphs for every integer s ≥ 1; here, the graph sP₃ denotes the disjoint union of s paths on three vertices. Our results show that both problems exhibit the same behaviour on H-free graphs (subject to some open cases). This is in part explained by a new general algorithm we design for finding in a graph G a largest induced subgraph whose blocks belong to some finite class C of graphs. We also compare our results with the state-of-the-art results for the Odd Cycle Transversal problem, which is known to behave differently on H-free graphs.

Abstract We resolve the computational complexity of Graph Isomorphism for classes of graphs characterized by two forbidden induced subgraphs $$ H_{1} $$ H 1 and $$H_2$$ H 2 for all but six pairs $$(H_1,H_2)$$ ( H 1 , H 2 ) . Schweitzer had previously shown that the number of open cases was finite, but without specifying the open cases. Grohe and Schweitzer proved that Graph Isomorphism is polynomial-time solvable on graph classes of bounded clique-width. Our work combines known results such as these with new results. By exploiting a relationship between Graph Isomorphism and clique-width, we simultaneously reduce the number of open cases for boundedness of clique-width for $$(H_1,H_2)$$ ( H 1 , H 2 ) -free graphs to five.

Tree-width and its linear variant path-width play a central role for the graph minor relation. In particular, Robertson and Seymour [ J. Combin. Theory Ser. B, 35 (1983), pp. 39--61] proved that for every tree $T$, the class of graphs that do not contain $T$ as a minor has bounded path-width. For the pivot-minor relation, rank-width and linear rank-width take over the role of tree-width and path-width. As such, it is natural to examine if, for every tree $T$, the class of graphs that do not contain $T$ as a pivot-minor has bounded linear rank-width. We first prove that this statement is false whenever $T$ is a tree that is not a caterpillar. We conjecture that the statement is true if $T$ is a caterpillar. We are also able to give partial confirmation of this conjecture by proving for every tree $T$, the class of $T$-pivot-minor-free distance-hereditary graphs has bounded linear rank-width if and only if $T$ is a caterpillar; for every caterpillar $T$ on at most four vertices, the class of $T$-pivot-minor-free graphs has bounded linear rank-width. To prove our second result, we only need to consider $T=P_4$ and $T=K_{1,3}$, but we follow a general strategy: first we show that the class of $T$-pivot-minor-free graphs is contained in some class of $(H_1,H_2)$-free graphs, which we then show to have bounded linear rank-width. In particular, we prove that the class of $(K_3,S_{1,2,2})$-free graphs has bounded linear rank-width, which strengthens a known result that this graph class has bounded rank-width.

A k-colouring c of a graph G is a mapping V(G) → {1,2,… k} such that c(u) ≠ c(v) whenever u and v are adjacent. The corresponding decision problem is Colouring. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as L(1,1)-Labelling). A classical complexity result on Colouring is a well-known dichotomy for H-free graphs, which was established twenty years ago (in this context, a graph is H-free if and only if it does not contain H as an induced subgraph). Moreover, this result has led to a large collection of results, which helped us to better understand the complexity of Colouring. In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years. We initiate such a systematic complexity study, and similar to the study of Colouring we use the class of H-free graphs as a testbed. We prove the following results: 1) We give almost complete classifications for the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring for H-free graphs. 2) If the number of colours k is fixed, that is, not part of the input, we give full complexity classifications for each of the three problems for H-free graphs. From our study we conclude that for fixed k the three problems behave in the same way, but this is no longer true if k is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs.

The Path Contraction problem has as input a graph G and an integer k and is to decide if G can be modified to the k-vertex path P_k by a sequence of edge contractions. A graph G is H-free for some graph H if G does not contain H as an induced subgraph. The Path Contraction problem restricted to H-free graphs is known to be NP-complete if H = claw or H = P₆ and polynomial-time solvable if H = P₅. We first settle the complexity of Path Contraction on H-free graphs for every H by developing a common technique. We then compare our classification with a (new) classification of the complexity of the problem Long Induced Path, which is to decide for a given integer k, if a given graph can be modified to P_k by a sequence of vertex deletions. Finally, we prove that the complexity classifications of Path Contraction and Cycle Contraction for H-free graphs do not coincide. The latter problem, which has not been fully classified for H-free graphs yet, is to decide if for some given integer k, a given graph contains the k-vertex cycle C_k as a contraction.

A large number of NP-hard graph problems are solvable in XP time when parameterized by some width parameter. Hence, when solving problems on special graph classes, it is helpful to know if the graph class under consideration has bounded width. In this paper we consider mim-width, a particularly general width parameter that has a number of algorithmic applications whenever a decomposition is "quickly computable" for the graph class under consideration. We start by extending the toolkit for proving (un)boundedness of mim-width of graph classes. By combining our new techniques with known ones we then initiate a systematic study into bounding mim-width from the perspective of hereditary graph classes, and make a comparison with clique-width, a more restrictive width parameter that has been well studied. We prove that for a given graph H, the class of H-free graphs has bounded mim-width if and only if it has bounded clique-width. We show that the same is not true for (H₁,H₂)-free graphs. We identify several general classes of (H₁,H₂)-free graphs having unbounded clique-width, but bounded mim-width, illustrating the power of mim-width. Moreover, we show that a branch decomposition of constant mim-width can be found in polynomial time, for these classes. Hence, as mentioned, these results have algorithmic implications: when the input is restricted to such a class of (H₁,H₂)-free graphs, many problems become polynomial-time solvable, including classical problems such as k-Colouring and Independent Set, domination-type problems known as LC-VSVP problems, and distance versions of LC-VSVP problems, to name just a few. We also prove a number of new results showing that, for certain H₁ and H₂, the class of (H₁,H₂)-free graphs has unbounded mim-width. Boundedness of clique-width implies boundedness of mim-width. By combining our results, which give both new bounded and unbounded cases for mim-width, with the known bounded cases for clique-width, we present summary theorems of the current state of the art for the boundedness of mim-width for (H₁,H₂)-free graphs. In particular, we classify the mim-width of (H₁,H₂)-free graphs for all pairs (H₁,H₂) with |V(H₁)| + |V(H₂)| ≤ 8. When H₁ and H₂ are connected graphs, we classify all pairs (H₁,H₂) except for one remaining infinite family and a few isolated cases.

Abstract A graph isH-free if it contains no induced subgraph isomorphic to H. We prove new complexity results for the two classical cycle transversal problemsFeedback Vertex SetandOdd Cycle Transversalby showing that they can be solved in polynomial time on $$(sP_1+ P_3)$$ (sP1+P3) -free graphs for every integer $$s\ge 1$$ s≥1 . We show the same result for the variantsConnected Feedback Vertex SetandConnected Odd Cycle Transversal. We also prove that the latter two problems are polynomial-time solvable on cographs; this was already known forFeedback Vertex SetandOdd Cycle Transversal. We complement these results by proving thatOdd Cycle TransversalandConnected Odd Cycle Transversalare -complete on $$(P_2+ P_5,P_6)$$ (P2+P5,P6) -free graphs.

Abstract The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list $$L(u)\subseteq \{1,\ldots ,k\},$$ L ( u ) ⊆ { 1 , … , k } , then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. The graph $$P_r+P_s$$ P r + P s is the disjoint union of the r-vertex path $$P_r$$ P r and the s-vertex path $$P_s.$$ P s . We prove that List 3-Colouring is polynomial-time solvable for $$(P_2+P_5)$$ ( P 2 + P 5 ) -free graphs and for $$(P_3+P_4)$$ ( P 3 + P 4 ) -free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices.

We obtain a complete classification of graphs H for which the class of (triangle,H)-free graphs is well-quasi-ordered by the induced subgraph relation and an almost complete classification of graphs H for which the class of (triangle,H)-free graphs has bounded clique-width. In particular, we show that for these graph classes, well-quasi-orderability implies boundedness of clique-width. To obtain our results, we further refine a known method based on canonical decomposition. This leads to a new decomposition technique that is applicable to both notions, well-quasi-orderability and clique-width.

A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The corresponding decision problem is called Disconnected Cut . This problem is known to be NP -hard on general graphs. We prove that it is polynomial-time solvable on claw-free graphs, answering a question of Ito et al. (TCS 2011). The basis for our result is a decomposition theorem for claw-free graphs of diameter 2, which we believe is of independent interest and builds on the research line initiated by Chudnovsky and Seymour (JCTB 2007–2012) and Hermelin et al. (ICALP 2011). On our way to exploit this decomposition theorem, we characterize how disconnected cuts interact with certain cobipartite subgraphs, and prove two further algorithmic results, namely that Disconnected Cut is polynomial-time solvable on circular-arc graphs and line graphs.

Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set ${\cal H}$ of forbidden induced subgraphs. We study the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the $|{\cal H}|=1$ case by classifying the boundedness of clique-width for every set ${\cal H}$ of self-complementary graphs. We then completely settle the $|{\cal H}|=2$ case. In particular, we determine one new class of $(H,\overline{H})$-free graphs of bounded clique-width (as a side effect, this leaves only five classes of $(H_1,H_2)$-free graphs, for which it is not known whether their clique-width is bounded). Once we have obtained the classification of the $|{\cal H}|=2$ case, we research the effect of forbidding self-complementary graphs on the boundedness of clique-width. Surprisingly, we show that for every set ${\cal F}$ of self-complementary graphs on at least five vertices, the classification of the boundedness of clique-width for $(\{H,\overline{H}\}\cup {\cal F})$-free graphs coincides with the one for the $|{\cal H}|=2$ case if and only if ${\cal F}$ does not include the bull.

A partition $(V_1,\ldots,V_k)$ of the vertex set of a graph $G$ with a (not necessarily proper) colouring $c$ is colourful if no two vertices in any $V_i$ have the same colour and every set $V_i$ induces a connected graph. The COLOURFUL PARTITION problem is to decide whether a coloured graph $(G,c)$ has a colourful partition of size at most $k$. This problem is closely related to the COLOURFUL COMPONENTS problem, which is to decide whether a graph can be modified into a graph whose connected components form a colourful partition by deleting at most $p$ edges. Nevertheless we show that COLOURFUL PARTITION and COLOURFUL COMPONENTS may have different complexities for restricted instances. We tighten known NP-hardness results for both problems and in addition we prove new hardness and tractability results for COLOURFUL PARTITION. Using these results we complete our paper with a thorough parameterized study of COLOURFUL PARTITION.

The Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for an integer k, such that no two adjacent vertices are coloured alike. A graph G is H-free if G does not contain H as an induced subgraph. It is known that Colouring is NP-complete for H-free graphs if H contains a cycle or claw, even for fixed k >= 3. We examine to what extent the situation may change if in addition the input graph has bounded diameter.

We introduce in a general setting a dynamic programming method for solving reconfiguration problems. Our method is based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge contractions that decrease the graph size without losing any critical information needed to solve the reconfiguration problem under consideration. Our general framework captures the approach behind known reconfiguration results of Bonsma (Discrete Appl Math 231:95–112, 2017) and Hatanaka et al. (IEICE Trans Fundam Electron Commun Comput Sci 98(6):1168–1178, 2015). As a third example, we apply the method to the following well-studied problem: given two k-colorings $$\alpha $$ and $$\beta $$ of a graph G, can $$\alpha $$ be modified into $$\beta $$ by recoloring one vertex of G at a time, while maintaining a k-coloring throughout? This problem is known to be PSPACE-hard even for bipartite planar graphs and $$k=4$$ . By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial-time algorithm for $$(k-2)$$ -connected chordal graphs.

The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer $$k\ge 0$$ , to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the 3-Colouring problem, or equivalently, to the problem of deciding whether or not a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for H-free graphs. We prove that it is NP-complete if H contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for $$P_4$$ -free graphs. We show that it remains polynomial-time solvable for $$P_5$$ -free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks whether or not a graph has an independent odd cycle transversal of size at most k for a given integer $$k\ge 0$$ . Finally, in line with our underlying research aim, we compare the complexity of Independent Feedback Vertex Set for H-free graphs with the complexity of 3-Colouring, Independent Odd Cycle Transversal and other related problems.

The $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for a fixed integer $k$ such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list $L(u) \subseteq \{1,\cdots, k\}$, then we obtain the List $k$-Colouring problem. A graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. We continue an extensive study into the complexity of these two problems for $H$-free graphs. The graph $P_r+P_s$ is the disjoint union of the $r$-vertex path $P_r$ and the $s$-vertex path $P_s$. We prove that List $3$-Colouring is polynomial-time solvable for $(P_2+P_5)$-free graphs and for $(P_3+P_4)$-free graphs. Combining our results with known results yields complete complexity classifications of $3$-Colouring and List $3$-Colouring on $H$-free graphs for all graphs $H$ up to seven vertices.

Let vc(G), fvs(G) and oct(G) denote, respectively, the size of a minimum vertex cover, minimum feedback vertex set and minimum odd cycle transversal in a graph G. One can ask, when looking for these sets in a graph, how much bigger might they be if we require that they are independent; that is, what is the price of independence? If G has a vertex cover, feedback vertex set or odd cycle transversal that is an independent set, then we let, respectively, ivc(G), ifvs(G) or ioct(G) denote the minimum size of such a set. We investigate for which graphs H the values of ivc(G), ifvs(G) and ioct(G) are bounded in terms of vc(G), fvs(G) and oct(G), respectively, when the graph G belongs to the class of H-free graphs. We find complete classifications for vertex cover and feedback vertex set and an almost complete classification for odd cycle transversal (subject to three non-equivalent open cases).

The Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a given integer k such that no two adjacent vertices are coloured alike. The complexity of Colouring is fully understood for graph classes characterized by one forbidden induced subgraph H. Despite a huge body of existing work, there are still major complexity gaps if two induced subgraphs H_1 and H_2 are forbidden. We let H_1 be the s-vertex cycle C_s and H_2 be the t-vertex path P_t. We show that Colouring is polynomial-time solvable for s=4 and t<=6, which unifies several known results for Colouring on (H_1,H_2)-free graphs. Our algorithm is based on a novel decomposition theorem for (C_4,P_6)-free graphs without clique cutsets into homogeneous pairs of sets and a new framework for bounding the clique-width of a graph by the clique-width of its subgraphs induced by homogeneous pairs of sets. To apply this framework, we also need to use divide-and-conquer to bound the clique-width of subgraphs induced by homogeneous pairs of sets. To complement our positive result we also prove that Colouring is NP-complete for s=4 and t>=9, which is the first hardness result on Colouring for (C_4,P_t)-free graphs.

The Near-Bipartiteness problem is that of deciding whether or not the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a forest. The set A in such a partition is said to be an independent feedback vertex set. Yang and Yuan proved that Near-Bipartiteness is polynomial-time solvable for graphs of diameter 2 and NP-complete for graphs of diameter 4. We show that Near-Bipartiteness is NP-complete for graphs of diameter 3, resolving their open problem. We also generalise their result for diameter 2 by proving that even the problem of computing a minimum independent feedback vertex is polynomial-time solvable for graphs of diameter 2.

The Colouring problem asks whether the vertices of a graph can be coloured with at most k colours for a given integer k in such a way that no two adjacent vertices receive the same colour. A graph is ( H 1 , H 2 ) -free if it has no induced subgraph isomorphic to H 1 or H 2 . A connected graph H 1 is almost classified if Colouring on ( H 1 , H 2 ) -free graphs is known to be polynomial-time solvable or NP -complete for all but finitely many connected graphs H 2 . We show that every connected graph H 1 apart from the claw K 1 , 3 and the 5-vertex path P 5 is almost classified. We also prove a number of new hardness results for Colouring on ( 2 P 2 , H ) -free graphs. This enables us to list all graphs H for which the complexity of Colouring is open on ( 2 P 2 , H ) -free graphs and all graphs H for which the complexity of Colouring is open on ( P 5 , H ) -free graphs. In fact we show that these two lists coincide. Moreover, we show that the complexities of Colouring for ( 2 P 2 , H ) -free graphs and for ( P 5 , H ) -free graphs are the same for all known cases.

A graph H is a square root of a graph G, or equivalently, G is the square of H, if G can be obtained from H by adding an edge between any two vertices in H that are of distance 2. The Square Root problem is that of deciding whether a given graph admits a square root. The problem of testing whether a graph admits a square root which belongs to some specified graph class $\mathcal {H}$ is called the $\mathcal {H}$ -Square Root problem. By showing boundedness of treewidth we prove that Square Root is polynomial-time solvable on some classes of graphs with small clique number and that $\mathcal {H}$ -Square Root is polynomial-time solvable when $\mathcal {H}$ is the class of cactuses.

A homomorphism from a graph G to a graph H is a vertex mapping f from the vertex set of G to the vertex set of H such that there is an edge between vertices f(u) and f(v) of H whenever there is an edge between vertices u and v of G. The H-Colouring problem is to decide whether or not a graph G allows a homomorphism to a fixed graph H. We continue a study on a variant of this problem, namely the Surjective $$H$$ -Colouring problem, which imposes the homomorphism to be vertex-surjective. We build upon previous results and show that this problem is NP-complete for every connected graph H that has exactly two vertices with a self-loop as long as these two vertices are not adjacent. As a result, we can classify the computational complexity of Surjective $$H$$ -Colouring for every graph H on at most four vertices.

The NP-complete problem Feedback Vertex Set is to decide if it is possible, for a given integer k>=0, to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the 3-Colouring problem, or equivalently, to the problem of deciding if a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for H-free graphs. We prove that it is NP-complete if H contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for P_4-free graphs. We show that it remains in P for P_5-free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks if a graph has an independent odd cycle transversal of size at most k for a given integer k>=0.

Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set H of forbidden induced subgraphs. We initiate a systematic study into the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the |H|=1 case by classifying the boundedness of clique-width for every set H of self-complementary graphs. We then completely settle the |H|=2 case. In particular, we determine one new class of (H1, complement of H1)-free graphs of bounded clique-width (as a side effect, this leaves only six classes of (H1, H2)-free graphs, for which it is not known whether their clique-width is bounded). Once we have obtained the classification of the |H|=2 case, we research the effect of forbidding self-complementary graphs on the boundedness of clique-width. Surprisingly, we show that for a set F of self-complementary graphs on at least five vertices, the classification of the boundedness of clique-width for ({H1, complement of H1} + F)-free graphs coincides with the one for the |H|=2 case if and only if F does not include the bull (the only non-empty self-complementary graphs on fewer than five vertices are P_1 and P_4, and P_4-free graphs have clique-width at most 2). Finally, we discuss the consequences of our results for COLOURING.

We continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree.

The Colouring problem is that of deciding, given a graph G and an integer k, whether G admits a (proper) k-colouring. For all graphs H up to five vertices, we classify the computational complexity of Colouring for (diamond,H)-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for (diamond,P1+2P2)-free graphs. Our technique for handling this case is to reduce the graph under consideration to a k-partite graph that has a very specific decomposition. As a by-product of this general technique we are also able to prove boundedness of clique-width for four other new classes of (H1,H2)-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of (H1,H2)-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8.

We consider the following graph modification problem. Let the input consist of a graph G = ( V , E ) , a weight function w : V ∪ E → N , a cost function c : V ∪ E → N 0 and a degree function δ : V → N 0 , together with three integers k v , k e and C . The question is whether we can delete a set of vertices of total weight at most k v and a set of edges of total weight at most k e so that the total cost of the deleted elements is at most C and every non-deleted vertex v has degree δ ( v ) in the resulting graph G ′ . We also consider the variant in which G ′ must be connected. Both problems are known to be NP -complete and W [ 1 ] -hard when parameterized by k v + k e . We prove that, when restricted to planar graphs, they stay NP -complete but have polynomial kernels when parameterized by k v + k e .

A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are at distance 2 from each other. The Square Root problem is that of deciding whether a given graph admits a square root. We consider this problem for planar graphs in the context of the “distance from triviality” framework. For an integer k , a planar + k v graph (or k -apex graph) is a graph that can be made planar by the removal of at most k vertices. We prove that a generalization of Square Root , in which some edges are prescribed to be either in or out of any solution, has a kernel of size O ( k ) for planar + k v graphs, when parameterized by k . Our result is based on a new edge reduction rule which, as we shall also show, has a wider applicability for the Square Root problem.

The class of H-free graphs has bounded clique-width if and only if H is an induced subgraph of the 4-vertex path P4. We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs H1 and H2. Prior to our study, it was not known whether the number of open cases was finite. We provide a positive answer to this question. To reduce the number of open cases, we determine new graph classes of bounded clique-width and new graph classes of unbounded clique-width. For obtaining the latter results, we first present a new, generic construction for graph classes of unbounded clique-width. Our results settle the boundedness or unboundedness of the clique-width of the class of (H1,H2)-free graphs for all pairs (H1,H2), both of which are connected, except two non-equivalent cases, and for all pairs (H1,H2), at least one of which is not connected, except 11 non-equivalent cases. We also consider classes characterized by forbidding a finite family of graphs {H1,…,Hp} as subgraphs, minors and topological minors, respectively, and completely determine which of these classes have bounded clique-width. Finally, we show algorithmic consequences of our results for the graph colouring problem restricted to (H1,H2)-free graphs.

The problem of finding a disconnected cut in a graph is NP‐hard in general but polynomial‐time solvable on planar graphs. The problem of finding a minimal disconnected cut is also NP‐hard but its computational complexity was not known for planar graphs. We show that it is polynomial‐time solvable on 3‐connected planar graphs but NP‐hard for 2‐connected planar graphs. Our technique for the first result is based on a structural characterization of minimal disconnected cuts in 3‐connected ‐free‐minor graphs and on solving a topological minor problem in the dual. In addition we show that the problem of finding a minimal connected cut of size at least 3 is NP‐hard for 2‐connected apex graphs. Finally, we relax the notion of minimality and prove that the problem of finding a so‐called semi‐minimal disconnected cut is still polynomial‐time solvable on planar graphs. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(4), 250–259 2016

For a positive integer |$k$| and graph |$G=(V,E)$|⁠, a |$k$|-colouring of |$G$| is a mapping |$c: V\rightarrow \{1,2,\ldots ,k\}$| such that |$c(u)\neq c(v)$| whenever |$uv\in E$|⁠. The |$k$|-Colouring problem is to decide, for a given |$G$|⁠, whether a |$k$|-colouring of |$G$| exists. The |$k$|-Precolouring Extension problem is to decide, for a given |$G=(V,E)$|⁠, whether a colouring of a subset of |$V$| can be extended to a |$k$|-colouring of |$G$|⁠. A |$k$|-list assignment of a graph is an allocation of a list—a subset of |$\{1,\ldots ,k\}$|—to each vertex, and the List |$k$|-Colouring problem is to decide, for a given |$G$|⁠, whether |$G$| has a |$k$|-colouring in which each vertex is coloured with a colour from its list. We consider the computational complexity of these three decision problems when restricted to graphs that do not contain a cycle on |$s$| vertices or a path on |$t$| vertices as induced subgraphs (for fixed positive integers |$s$| and |$t$|⁠). We report on past work and prove a number of new NP-completeness results.

Paths $P_1,\ldots,P_k$ in a graph $G=(V,E)$ are said to be mutually induced if for any $1\leq i

We investigate the problem of modifying a graph into a connected graph in which the degree of each vertex satisfies a prescribed parity constraint. Let ea, ed and vd denote the operations edge addition, edge deletion and vertex deletion respectively. For any S subseteq {ea,ed,vd}, we define Connected Degree Parity Editing (S) (CDPE(S)) to be the problem that takes as input a graph G, an integer k and a function delta: V(G) -> {0,1}, and asks whether G can be modified into a connected graph H with d_H(v) = delta(v)(mod 2) for each v in V(H), using at most k operations from S. We prove that (*) if S={ea} or S={ea,ed}, then CDPE(S) can be solved in polynomial time; (*) if {vd} subseteq S subseteq {ea,ed,vd}, then CDPE(S) is NP-complete and W-hard when parameterized by k, even if delta = 0. Together with known results by Cai and Yang and by Cygan, Marx, Pilipczuk, Pilipczuk and Schlotter, our results completely classify the classical and parameterized complexity of the CDPE(S) problem for all S subseteq {ea,ed,vd}. We obtain the same classification for a natural variant of the cdpe(S) problem on directed graphs, where the target is a weakly connected digraph in which the difference between the in- and out-degree of every vertex equals a prescribed value. As an important implication of our results, we obtain polynomial-time algorithms for Eulerian Editing problem and its directed variant. To the best of our knowledge, the only other natural non-trivial graph class H for which the H-Editing problem is known to be polynomial-time solvable is the class of split graphs.

For a set S of graphs, a perfect S -packing ( S -factor) of a graph G is a set of mutually vertex-disjoint subgraphs of G that each are isomorphic to a member of S and that together contain all vertices of G . If G allows a covering (locally bijective homomorphism) to a graph H , i.e., a vertex mapping f : V G → V H satisfying the property that f ( u ) f ( v ) belongs to E H whenever the edge u v belongs to E G such that for every u ∈ V G the restriction of f to the neighborhood of u is bijective, then G is an H -cover. For some fixed H let S ( H ) consist of all connected H -covers. Let K k , ℓ be the complete bipartite graph with partition classes of size k and ℓ , respectively. For all fixed k , ℓ ≥ 1 , we determine the computational complexity of the problem that tests whether a given bipartite graph has a perfect S ( K k , ℓ ) -packing. Our technique is partially based on exploring a close relationship to pseudo-coverings. A pseudo-covering from a graph G to a graph H is a homomorphism from G to H that becomes a covering to H when restricted to a spanning subgraph of G . We settle the computational complexity of the problem that asks whether a graph allows a pseudo-covering to K k , ℓ for all fixed k , ℓ ≥ 1 .

The modular treewidth of a graph is its treewidth after the contraction of modules. Modular treewidth properly generalizes treewidth and is itself properly generalized by clique-width. We show that the number of satisfying assignments of a CNF formula whose incidence graph has bounded modular treewidth can be computed in polynomial time. This provides new tractable classes of formulas for which #SAT is polynomial. In particular, our result generalizes known results for the treewidth of incidence graphs and is incomparable with known results for clique-width (or rank-width) of signed incidence graphs. The contraction of modules is an effective data reduction procedure. Our algorithm is the first one to harness this technique for #SAT. The order of the polynomial time bound of our algorithm depends on the modular treewidth. We show that this dependency cannot be avoided subject to an assumption from Parameterized Complexity.

The k-in-a-Path problem is to test whether a graph contains an induced path spanning k given vertices. This problem is NP-complete in general graphs, already when k=3. We show how to solve it in polynomial time on claw-free graphs, when k is an arbitrary fixed integer not part of the input. As a consequence, also the k-Induced Disjoint Paths and the k-in-a-Cycle problem are solvable in polynomial time on claw-free graphs for any fixed k. The first problem has as input a graph G and k pairs of specified vertices (s i ,t i ) for i=1,…,k and is to test whether G contain k mutually induced paths P i such that P i connects s i and t i for i=1,…,k. The second problem is to test whether a graph contains an induced cycle spanning k given vertices. When k is part of the input, we show that all three problems are NP-complete, even for the class of line graphs, which form a subclass of the class of claw-free graphs.

Abstract We discuss 11 known basic models of distributed computing: four message‐passing models that differ by the (non)existence of port‐numbers and a hierarchy of seven local computations models. In each of these models, we study the computational complexity of the decision problems if the leader election and if the naming problem can be solved on a given network. It is already known that these two decision problems are solvable in polynomial time for two models and are co‐NP‐complete for another one. Here, we settle the computational complexity for both problems in the remaining eight models by showing that they are co‐NP‐complete. We do this by translating each problem into a graph labeling problem. By using this technique, we also obtain an alternative proof for the already known co‐NP‐completeness result. In the second part of our article, we completely classify the computational complexity of all the corresponding graph labeling problems, i.e., for every fixed integer $k\geq 1$ we determine the complexity of the problem that asks whether a given graph allows a certain graph labeling that uses at most k labels. We also explain the close relationship of these labelings to graph homomorphisms that satisfy some further (global or local) constraints. This yields a new class of “constrained” graph homomorphisms that include the already known locally constrained graph homomorphisms. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011

We study the propositional satisfiability problem (SAT) on classes of CNF formulas (formulas in Conjunctive Normal Form) that obey certain structural restrictions in terms of their hypergraph structure, by associating to a CNF formula the hypergraph obtained by ignoring negations and considering clauses as hyperedges on variables. We show that satisfiability of CNF formulas with so-called ``beta-acyclic hypergraphs'' can be decided in polynomial time. We also study the parameterized complexity of SAT for ``almost'' beta-acyclic instances, using as parameter the formula's distance from being beta-acyclic. As distance we use the size of smallest strong backdoor sets and the beta-hypertree width. As a by-product we obtain the W[1]-hardness of SAT parameterized by the (undirected) clique-width of the incidence graph, which disproves a conjecture by Fischer, Makowsky, and Ravve (Discr. Appl. Math. 156, 2008).

Abstract For a fixed pattern graph H, let H‐CONTRACTIBILITY denote the problem of deciding whether a given input graph is contractible to H. This paper is part I of our study on the computational complexity of the H‐CONTRACTIBILITY problem. We continue a line of research that was started in 1987 by Brouwer and Veldman, and we determine the computational complexity of the H‐CONTRACTIBILITY problem for certain classes of pattern graphs. In particular, we pinpoint the complexity for all graphs H with five vertices except for two graphs, whose polynomial time algorithms are presented in part II. Interestingly, in all connected cases that are known to be polynomially solvable, the pattern graph H has a dominating vertex, whereas in all cases that are known to be NP‐complete, the pattern graph H does not have a dominating vertex. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008

Abstract For a fixed pattern graph H, let H‐CONTRACTIBILITY denote the problem of deciding whether a given input graph is contractible to H. This article is part II of our study on the computational complexity of the H‐CONTRACTIBILITY problem. In the first article we pinpointed the complexity for all pattern graphs with five vertices except for two pattern graphs H. Here, we present polynomial time algorithms for these two remaining pattern graphs. Interestingly, in all connected cases that are known to be polynomially solvable, the pattern graph H has a dominating vertex, whereas in all cases that are known to be NP‐complete, the pattern graph H does not have a dominating vertex. © 2008 Wiley Periodicals, Inc. NETWORKS, 2008

We first show that for any bipartite graph H with at most five vertices there exists an on-line competitive algorithm for the class of H-free bipartite graphs. We then analyze the performance of an on-line algorithm for coloring bipartite graphs on various subfamilies. The algorithm yields new upper bounds for the on-line chromatic number of bipartite graphs. We prove that the algorithm is on-line competitive for $P_7$-free bipartite graphs, i.e., that do not contain an induced path on seven vertices. The number of colors used by the on-line algorithm for $P_6$-free and $P_7$-free bipartite graphs is, respectively, bounded by roughly twice and roughly eight times the on-line chromatic number. In contrast, it is known that there exists no competitive on-line algorithm to color $P_6$-free (or $P_7$-free) bipartite graphs, i.e., for which the number of colors is bounded by any function depending only on the chromatic number.

This work evaluates an algorithm that maps a number of communicating processes to a heterogeneous tiled system on chip (SoC) architecture at run-time. The mapping algorithm minimizes the total amount of energy consumption, while still providing an adequate quality of service (QoS). A realistic example is mapped using this algorithm.