The Upper Clique Transversal (UCT) problem asks for the size of the largest minimal set of vertices intersecting all the maximal cliques of the input graph. This problem was recently introduced by Milanič and Uno [WG 2023], who studied its complexity on several graph classes. They showed that the problem is NP-hard on chordal graphs, and gave polynomial-time algorithms for UCT on split and on proper interval graphs. They left open the complexity of UCT on interval graphs. In this work we settle this question by giving a polynomial-time algorithm for UCT on interval graphs. We show that even on the more general class of rooted directed path graphs, which can be understood as a "tree-like version" of interval graphs, the problem remains polynomial-time solvable. On the negative side, we observe as consequences of the NP-hardness proof for chordal graphs due to Milanič and Uno that the problem is NP-hard on graphs of path-independence number two (interval graphs have path-independence number one) and on well-partitioned chordal graphs which lie between split and chordal graphs.

We study a generalization of the classic Global Min-Cut problem, called Global Label Min-Cut (or sometimes Global Hedge Min-Cut ): the edges of the input (multi)graph are labeled (or partitioned into color classes or hedges), and removing all edges of the same label (color or from the same hedge) costs one. The problem asks to disconnect the graph at minimum cost. While the \(st\) -cut version of the problem is known to be \(\mathsf{NP}\) -hard, the above global cut version is known to admit a quasi-polynomial randomized \(n^{\mathcal{O}(\log\mathrm{OPT})}\) -time algorithm due to Ghaffari, Karger, and Panigrahi [SODA 2017]. They consider this as “strong evidence that this problem is in P ” . We show that this is actually not the case. We complete the study of the complexity of the Global Label Min-Cut problem by showing that the quasi-polynomial running time is probably optimal: We show that the existence of an algorithm with running time \((np)^{o(\log n/(\log\log n)^{2})}\) would contradict the Exponential Time Hypothesis, where \(n\) is the number of vertices, and \(p\) is the number of labels in the input. The key step for the lower bound is a proof that Global Label Min-Cut is \(\mathsf{W}\) [1]-hard when parameterized by the number of uncut labels . In other words, the problem is difficult in the regime where almost all labels need to be cut to disconnect the graph.

We prove that Hamiltonian Path and Hamiltonian Cycle are NP -hard on graphs of linear mim-width 26, even when a linear order of the input graph with mim-width 26 is provided together with input. This fills a gap left by a broken proof of the para-NP-hardness of Hamiltonicity problems parameterized by mim-width.

F-branchwidth is a framework for width measures of graphs, recently introduced by Eiben et al. [ITCS 2022], that captures tree-width, co-tree-width, clique-width, and mim-width, and several of their generalizations and interpolations. In this work, we search for algorithmic applications of F-branchwidth measures that do not have an equivalent counterpart in the literature so far. Our first contribution is a minimal set of eleven F-branchwidth measures such that each of the infinitely many F-branchwidth measures is equivalent to one of the eleven. We observe that for the FO Model Checking problem, each F-branchwidth is either equivalent to clique-width (and therefore has an FPT-algorithm by formula length plus the width) or the problem remains as hard as on general graphs even on graphs of constant width. Next, we study the number of equivalence classes of the neighborhood equivalence in a decomposition, which upper bounds the run time of the model checking algorithm for ACDN logic recently introduced by Bergougnoux et al. [SODA 2023]. We give structural lower bounds that show that for each F-branchwidth, an efficient model checking algorithm was already known or cannot be obtained via this method. Lastly, we classify the complexity of Independent Set parameterized by any F-branchwidth except for one open case. Also here, our contributions are lower bounds. In this context, we also prove that Independent Set on graphs of mim-width w cannot be solved in time n^o(w) unless the Exponential Time Hypothesis fails, answering an open question in the literature.

We prove that Hamiltonian Path and Hamiltonian Cycle are NP-hard on graphs of linear mim-width 26, even when a linear order of the input graph with mim-width 26 is provided together with input. This fills a gap left by a broken proof of the para-NP-hardness of Hamiltonicity problems parameterized by mim-width.

Abstract In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna (ACM Trans Comput Theory 9:1–36, 2018), that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, ( q -)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, Capacitated Vertex Cover and Bipartite Bandwidth.

Abstract We initiate the study of theDiverse Pair of (Maximum/ Perfect) Matchingsproblems which given a graphGand an integerk, ask whetherGhas two (maximum/perfect) matchings whose symmetric difference is at leastk.Diverse Pair of Matchings(asking for two not necessarily maximum or perfect matchings) is $$\textsf{NP}$$ NP -complete on general graphs ifkis part of the input, and we consider two restricted variants. First, we show that on bipartite graphs, the problem is polynomial-time solvable, and second we show thatDiverse Pair of Maximum Matchingsis $$\textsf{FPT}$$ FPT parameterized byk. We round off the work by showing thatDiverse Pair of Matchingshas a kernel on $${\mathcal {O}}(k^2)$$ O(k2) vertices.

Abstract We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial time results on graph classes, and answers open questions posed by Campos and Silva (Algorithmica 80(1), 104–115, 2018) and Bonomo et al. (Graphs and Combinatorics 25(2), 153–167, 2009). This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is $$\textsf{FPT}$$ FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound. The running times of the clique-width based algorithms for $$b$$ b -Coloring and Fall Coloring are tight under the Exponential Time Hypothesis.

Abstract. We confirm a conjecture of Gartland and Lokshtanov [SODA 2023]: if for a hereditary graph class [Formula: see text] there exists a constant [Formula: see text] such that no member of [Formula: see text] contains a [Formula: see text]-creature as an induced subgraph or a [Formula: see text]-skinny-ladder as an induced minor, then there exists a polynomial [Formula: see text] such that every [Formula: see text] contains at most [Formula: see text] minimal separators. By a result of Fomin, Todinca, and Villanger [ SIAM J. Comput., 44 (2015), pp. 54–87] the latter entails the existence of polynomial-time algorithms for Maximum Weight Independent Set, Feedback Vertex Set and many other problems, when restricted to an input graph from [Formula: see text]. Furthermore, as shown by Gartland and Lokshtanov, our result implies a full dichotomy of hereditary graph classes defined by a finite set of forbidden induced subgraphs into tame (admitting a polynomial bound of the number of minimal separators) and feral (containing infinitely many graphs with exponential number of minimal separators).

The b-Coloring problem, which given a graph G and an integer k asks whether G has a proper k-coloring such that each color class has a vertex adjacent to all color classes except its own, is known to be FPT parameterized by the vertex cover number and XP and 𝖶[1]-hard parameterized by clique-width. Its complexity when parameterized by the treewidth of the input graph remained an open problem. We settle this question by showing that b-Coloring is XNLP-complete when parameterized by the pathwidth of the input graph. Besides determining the precise parameterized complexity of this problem, this implies that b-Coloring parameterized by pathwidth is 𝖶[t]-hard for all t, and resolves the parameterized complexity of b-Coloring parameterized by treewidth. We complement this result by showing that b-Coloring is FPT when parameterized by neighborhood diversity and by twin cover, two parameters that generalize vertex cover to more dense graphs, but are incomparable to pathwidth.

In this paper, we show that Treewidth is NP-complete for cubic graphs, thereby improving the result by Bodlaender and Thilikos from 1997 that Treewidth is NP-complete on graphs with maximum degree at most 9. We add a new and simpler proof of the NP-completeness of treewidth, and show that Treewidth remains NP-complete on subcubic induced subgraphs of the infinite 3-dimensional grid.

We revisit a graph width parameter that we dub bipartite treewidth, along with its associated graph decomposition that we call bipartite tree decomposition. Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one vertex from the bipartite part of any other bag, while the width of such decomposition measures how far the bags are from being bipartite. Adapted from a tree decomposition originally defined by Demaine, Hajiaghayi, and Kawarabayashi [SODA 2010] and explicitly defined by Tazari [Theor. Comput. Sci. 2012], bipartite treewidth appears to play a crucial role for solving problems related to odd-minors, which have recently attracted considerable attention. As a first step toward a theory for solving these problems efficiently, the main goal of this paper is to develop dynamic programming techniques to solve problems on graphs of small bipartite treewidth. For such graphs, we provide a number of para-NP-completeness results, FPT-algorithms, and XP-algorithms, as well as several open problems. In particular, we show that K_t-Subgraph-Cover, Weighted Vertex Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are FPT parameterized by bipartite treewidth. We also provide the following complexity dichotomy when H is a 2-connected graph, for each of the H-Subgraph-Packing, H-Induced-Packing, H-Scattered-Packing, and H-Odd-Minor-Packing problems: if H is bipartite, then the problem is para-NP-complete parameterized by bipartite treewidth while, if H is non-bipartite, then the problem is solvable in XP-time. Beyond bipartite treewidth, we define 1-ℋ-treewidth by replacing the bipartite graph class by any graph class ℋ. Most of the technology developed here also works for this more general parameter.

We introduce a logic called distance neighborhood logic with acyclicity and connectivity constraints (A&C DN for short) which extends existential MSO1 with predicates for querying neighborhoods of vertex sets in various powers of a graph and for verifying connectivity and acyclicity of vertex sets. Building upon [Bergougnoux and Kante, ESA 2019; SIDMA 2021], we show that the model checking problem for every fixed A&C DN formula is solvable in nO(w) time when the input graph is given together with a branch decomposition of mim-width W. Nearly all problems that are known to be solvable in polynomial time given a branch decomposition of constant mim-width can be expressed in this framework. We add several natural problems to this list, including problems asking for diverse sets of solutions. Our model checking algorithm is efficient whenever the given branch decomposition of the input graph has small index in terms of the d-neighborhood equivalence [Bui-Xuan, Telle, and Vatshelle, TCS 2013]. We therefore unify and extend known algorithms for tree-width, clique-width and rank-width. Our algorithm has a single-exponential dependence on these three width measures and asymptotically matches run times of the fastest known algorithms for several problems. This results in algorithms with tight run times under the Exponential Time Hypothesis (ETH) for tree-width, clique-width and rank-width; the above mentioned run time for mim-width is nearly tight under the ETH for several problems as well. Our results are also tight in terms of the expressive power of the logic: we show that already slight extensions of our logic make the model checking problem para-NP-hard when parameterized by mim-width plus formula length. * The full version of the paper can be accessed at https://arxiv.org/abs/2202.13335. This research is part of a project that has received funding from the Research Council of Norway Grant Agreement 274526 (LJ).

We show fixed-parameter tractability of the DIRECTED MULTICUT problem with three terminal pairs (with a randomized algorithm). In this problem we are given a directed graph G, three pairs of vertices (called terminals) (s1, t1), (s2, t2), (s3, t3), and an integer k and we want to find a set of at most k non-terminal vertices in G that intersect all s1t1-paths, all s2t2-paths, and all s3t3-paths. The parameterized complexity of this problem has been open since Chitnis, Hajiaghayi, and Marx proved fixed-parameter tractability of the two-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlström proved the W[1]-hardness of the four-terminal-pairs case at SODA 2016. On the technical side, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch, Pilipczuk, Wahlström, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain. We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomassé, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona de Mendez, Simon, Thomassé, Toruńczyk, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor, a vertex corresponding to the “middle” box in the grid minor can be proclaimed irrelevant — not contained in the sought solution — and thus reduced. * The full version of the paper can be accessed at https://arxiv.org/abs/2207.07425. The research leading to the results presented in this paper was partially carried out during the Parameterized Algorithms Retreat of the University of Warsaw, PARUW 2022, held in Bedlewo in April 2022. This research is a part of projects that have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme Grant Agreement 714704 (TM, MP) and 648527 (MH), from the Alexander von Humboldt Foundation (MS), from the Research Council of Norway (LJ), and by the Federal Ministry of Education and Research (BMBF) and by a fellowship within the IFI programme of the German Academic Exchange Service (DAAD). (MH).

We study a generalization of the classic GLOBAL MIN-CUT problem, called GLOBAL LABEL MIN-CUT (or sometimes GLOBAL HEDGE MIN-CUT): the edges of the input (multi)graph are labeled (or partitioned into color classes or hedges), and removing all edges of the same label (color or from the same hedge) costs one. The problem asks to disconnect the graph at minimum cost. While the st-cut version of the problem is known to be NP-hard, the above global cut version is known to admit a quasi-polynomial randomized nO(log OPT)-time algorithm due to Ghaffari, Karger, and Panigrahi [SODA 2017]. They consider this as “strong evidence that this problem is in P”. We show that this is actually not the case. We complete the study of the complexity of the Global Label Min-Cut problem by showing that the quasi-polynomial running time is probably optimal: We show that the existence of an algorithm with running time (np)o(log n/(log log n)2) would contradict the Randomized Exponential Time Hypothesis, where n is the number of vertices, and p is the number of labels in the input. The key step for the lower bound is a proof that Global Label Min-Cut is W[1]-hard when parameterized by the number of uncut labels. In other words, the problem is difficult in the regime where almost all labels need to be cut to disconnect the graph. To turn this lower bound into a quasi-polynomial-time lower bound, we also needed to revisit the framework due to Marx [Theory Comput. 2010] of proving lower bounds assuming Exponential Time Hypothesis through the SUBGRAPH ISOMORPHISM problem parameterized by the number of edges of the pattern. Here, we provide an alternative simplified proof of the hardness of this problem that is more versatile with respect to the choice of the regimes of the parameters. * This research is a part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme Grant Agreement 714704 (LJ, TM, MP, US) and from the Research Council of Norway (LJ).

The q-Coloring problem asks whether the vertices of a graph can be properly colored with q colors. In this paper we perform a fine-grained analysis of the complexity of q-Coloring with respect to a hierarchy of structural parameters. We show that unless the Exponential Time Hypothesis fails, there is no constant θ such that q-Coloring parameterized by the size k of a vertex cover can be solved in O∗(θk) time for all fixed q. We prove that there are O∗((q−ɛ)k) time algorithms where k is the vertex deletion distance to several graph classes for which q-Coloring is known to be solvable in polynomial time, including all graph classes F whose (q+1)-colorable members have bounded treedepth. In contrast, we prove that if F is the class of paths – some of the simplest graphs of unbounded treedepth – then no such algorithm can exist unless the Strong Exponential Time Hypothesis fails.

Abstract In this work, we give a structural lemma on merges of typical sequences, a notion that was introduced in 1991 [Lagergren and Arnborg, Bodlaender and Kloks, both ICALP 1991] to obtain constructive linear time parameterized algorithms for treewidth and pathwidth. The lemma addresses a runtime bottleneck in those algorithms but so far it does not lead to asymptotically faster algorithms. However, we apply the lemma to show that the cutwidth and the modified cutwidth of series parallel digraphs can be computed in polynomial time.

We confirm a conjecture of Gartland and Lokshtanov [arXiv:2007.08761]: if for a hereditary graph class 𝒢 there exists a constant k such that no member of 𝒢 contains a k-creature as an induced subgraph or a k-skinny-ladder as an induced minor, then there exists a polynomial p such that every G ∈ 𝒢 contains at most p(|V(G)|) minimal separators. By a result of Fomin, Todinca, and Villanger [SIAM J. Comput. 2015] the latter entails the existence of polynomial-time algorithms for Maximum Weight Independent Set, Feedback Vertex Set and many other problems, when restricted to an input graph from 𝒢. Furthermore, as shown by Gartland and Lokshtanov, our result implies a full dichotomy of hereditary graph classes defined by a finite set of forbidden induced subgraphs into tame (admitting a polynomial bound of the number of minimal separators) and feral (containing infinitely many graphs with exponential number of minimal separators).

Algorithms for computing or approximating optimal decompositions for decompositional parameters such as treewidth or clique-width have so far traditionally been tailored to specific width parameters. Moreover, for mim-width, no efficient algorithms for computing good decompositions were known, even under highly restrictive parameterizations. In this work we identify F-branchwidth as a class of generic decompositional parameters that can capture mim-width, treewidth, clique-width as well as other measures. We show that while there is an infinite number of F-branchwidth parameters, only a handful of these are asymptotically distinct. We then develop fixed-parameter and kernelization algorithms (under several structural parameterizations) that can compute every possible F-branchwidth, providing a unifying framework that can efficiently obtain near-optimal tree-decompositions, k-expressions, as well as optimal mim-width decompositions.

For nonempty σ, ρ ⊆ ℕ, a vertex set S in a graph G is a (σ, ρ)-dominating set if for all v ∈ S, |N(v) ∩ S| ∈ σ, and for all v ∈ V(G) ⧵ S, |N(v) ∩ S| ∈ ρ. The Min/Max (σ,ρ)-Dominating Set problems ask, given a graph G and an integer k, whether G contains a (σ, ρ)-dominating set of size at most k and at least k, respectively. This framework captures many well-studied graph problems related to independence and domination. Bui-Xuan, Telle, and Vatshelle [TCS 2013] showed that for finite or co-finite σ and ρ, the Min/Max (σ,ρ)-Dominating Set problems are solvable in XP time parameterized by the mim-width of a given branch decomposition of the input graph. In this work we consider the parameterized complexity of these problems and obtain the following: For minimization problems, we complete several scattered W[1]-hardness results in the literature to a full dichotomoy into polynomial-time solvable and W[1]-hard cases, and for maximization problems we obtain the same result under the additional restriction that σ and ρ are finite sets. All W[1]-hard cases hold assuming that a linear branch decomposition of bounded mim-width is given, and with the solution size being an additional part of the parameter. Furthermore, for all W[1]-hard cases we also rule out f(w)n^o(w/log w)-time algorithms assuming the Exponential Time Hypothesis, where f is any computable function, n is the number of vertices and w the mim-width of the given linear branch decomposition of the input graph.

In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, and Bipartite Bandwidth.

An intersection digraph is a digraph where every vertex $v$ is represented by an ordered pair $(S_v, T_v)$ of sets such that there is an edge from $v$ to $w$ if and only if $S_v$ and $T_w$ intersect. An intersection digraph is reflexive if $S_v\cap T_v\neq \emptyset$ for every vertex $v$. Compared to well-known undirected intersection graphs like interval graphs and permutation graphs, not many algorithmic applications on intersection digraphs have been developed. Motivated by the successful story on algorithmic applications of intersection graphs using a graph width parameter called mim-width, we introduce its directed analogue called `bi-mim-width' and prove that various classes of reflexive intersection digraphs have bounded bi-mim-width. In particular, we show that as a natural extension of $H$-graphs, reflexive $H$-digraphs have linear bi-mim-width at most $12|E(H)|$, which extends a bound on the linear mim-width of $H$-graphs [On the Tractability of Optimization Problems on $H$-Graphs. Algorithmica 2020]. For applications, we introduce a novel framework of directed versions of locally checkable problems, that streamlines the definitions and the study of many problems in the literature and facilitates their common algorithmic treatment. We obtain unified polynomial-time algorithms for these problems on digraphs of bounded bi-mim-width, when a branch decomposition is given. Locally checkable problems include Kernel, Dominating Set, and Directed $H$-Homomorphism.

Abstract A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed $$q \ge 2$$ q ≥ 2 , we give an $$\mathscr {O}^{\star }(q^{{\mathsf {tw}}})$$ O ⋆ ( q tw ) -time algorithm when the input graph is given together with one of its tree decompositions of width $${\mathsf {tw}} $$ tw . We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is $$\mathsf {XP}$$ XP parameterized by clique-width.

We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial-time results on graph classes, and answers open questions posed by Campos and Silva [Algorithmica, 2018] and Bonomo et al. [Graphs Combin., 2009]. This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound. The running times of the clique-width based algorithms for b-Coloring and Fall Coloring are tight under the Exponential Time Hypothesis.

When modeling an application of practical relevance as an instance of a combinatorial problem X, we are often interested not merely in finding one optimal solution for that instance, but in finding a sufficiently diverse collection of good solutions. In this work we initiate a systematic study of diversity from the point of view of fixed-parameter tractability theory. We consider an intuitive notion of diversity of a collection of solutions which suits a large variety of combinatorial problems of practical interest. Our main contribution is an algorithmic framework which --automatically-- converts a tree-decomposition-based dynamic programming algorithm for a given combinatorial problem X into a dynamic programming algorithm for the diverse version of X. Surprisingly, our algorithm has a polynomial dependence on the diversity parameter.

We initiate the study of the Diverse Pair of (Maximum/ Perfect) Matchings problems which given a graph G and an integer k, ask whether G has two (maximum/perfect) matchings whose symmetric difference is at least k. Diverse Pair of Matchings (asking for two not necessarily maximum or perfect matchings) is NP-complete on general graphs if k is part of the input, and we consider two restricted variants. First, we show that on bipartite graphs, the problem is polynomial-time solvable, and second we show that Diverse Pair of Maximum Matchings is FPT parameterized by k. We round off the work by showing that Diverse Pair of Matchings has a kernel on 𝒪(k²) vertices.

It can be shown that each permutation group G ⊑ 𝕊_n can be embedded, in a well defined sense, in a connected graph with O(n+|G|) vertices. Some groups, however, require much fewer vertices. For instance, 𝕊_n itself can be embedded in the n-clique K_n, a connected graph with n vertices. In this work, we show that the minimum size of a context-free grammar generating a finite permutation group G⊑ 𝕊_n can be upper bounded by three structural parameters of connected graphs embedding G: the number of vertices, the treewidth, and the maximum degree. More precisely, we show that any permutation group G ⊑ 𝕊_n that can be embedded into a connected graph with m vertices, treewidth k, and maximum degree Δ, can also be generated by a context-free grammar of size 2^{O(kΔlogΔ)}⋅ m^{O(k)}. By combining our upper bound with a connection established by Pesant, Quimper, Rousseau and Sellmann [Gilles Pesant et al., 2009] between the extension complexity of a permutation group and the grammar complexity of a formal language, we also get that these permutation groups can be represented by polytopes of extension complexity 2^{O(kΔlogΔ)}⋅ m^{O(k)}. The above upper bounds can be used to provide trade-offs between the index of permutation groups, and the number of vertices, treewidth and maximum degree of connected graphs embedding these groups. In particular, by combining our main result with a celebrated 2^{Ω(n)} lower bound on the grammar complexity of the symmetric group 𝕊_n due to Glaister and Shallit [Glaister and Shallit, 1996] we have that connected graphs of treewidth o(n/log n) and maximum degree o(n/log n) embedding subgroups of 𝕊_n of index 2^{cn} for some small constant c must have n^{ω(1)} vertices. This lower bound can be improved to exponential on graphs of treewidth n^{ε} for ε < 1 and maximum degree o(n/log n).

We initialize a series of papers deepening the understanding of algorithmic properties of the width parameter maximum induced matching width (mim-width) of graphs. In this first volume we provide the first polynomial-time algorithms on graphs of bounded mim-width for problems that are not locally checkable. In particular, we givenO(w)-time algorithms on graphs of mim-width at most w, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and H -Induced Topological Minor for fixed H. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Permutation and Circular Permutation graphs, Convex graphs, k -Trapezoid, Circular k -Trapezoid, k -Polygon, Dilworth-k and Co- k -Degenerate graphs for fixed k. We contrast these positive results to the fact that problems about finding long non-induced paths remain hard on graphs of bounded mim-width: We show that Hamiltonian Cycle (and hence Hamiltonian Path) is NP-hard on graphs of linear mim-width 1; this further hints at the expressive power of the mim-width parameter.

A b-coloring of a graph G is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The b-Coloring problem asks whether a graph G has a b-coloring with k colors. The b-chromatic number of a graph G, denoted by χb(G), is the maximum number k such that G admits a b-coloring with k colors. We consider the complexity of the b-Coloring problem, whenever the value of k is close to one of two upper bounds on χb(G): The maximum degree Δ(G) plus one, and the m-degree, denoted by m(G), which is defined as the maximum number i such that G has i vertices of degree at least i−1. We obtain a dichotomy result for all fixed k∈N when k is close to one of the two above mentioned upper bounds. Concretely, we show that if k∈{Δ(G)+1−p,m(G)−p}, the problem is polynomial-time solvable whenever p∈{0,1} and, even when k=3, it is NP-complete whenever p≥2. We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree Δ(G) of the input graph G and give two FPT-algorithms. First, we show that deciding whether a graph G has a b-coloring with m(G) colors is FPT parameterized by Δ(G). Second, we show that b-Coloring is FPT parameterized by Δ(G)+ℓk(G), where ℓk(G) denotes the number of vertices of degree at least k.

We generalize the family of (σ,ρ) problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as Distance-r Dominating Set and Distance-r Independent Set. We show that these distance problems are in XP parameterized by the structural parameter mim-width, and hence polynomial-time solvable on graph classes where mim-width is bounded and quickly computable, such as k-trapezoid graphs, Dilworth k-graphs, (circular) permutation graphs, interval graphs and their complements, convex graphs and their complements, k-polygon graphs, circular arc graphs, complements of d-degenerate graphs, and H-graphs if given an H-representation. We obtain these results by showing that taking any power of a graph never increases its mim-width by more than a factor of two. To supplement these findings, we show that many classes of (σ,ρ) problems are W[1]-hard parameterized by mim-width + solution size. We show that powers of graphs of tree-width w−1 or path-width w and powers of graphs of clique-width w have mim-width at most w. These results provide new classes of bounded mim-width. We prove a slight strengthening of the first statement which implies that, surprisingly, Leaf Power graphs which are of importance in the field of phylogenetic studies have mim-width at most 1.

We generalize the family of $(σ, ρ)$-problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as distance-$r$ dominating set and distance-$r$ independent set. We show that these distance problems are XP parameterized by the structural parameter mim-width, and hence polynomial on graph classes where mim-width is bounded and quickly computable, such as $k$-trapezoid graphs, Dilworth $k$-graphs, (circular) permutation graphs, interval graphs and their complements, convex graphs and their complements, $k$-polygon graphs, circular arc graphs, complements of $d$-degenerate graphs, and $H$-graphs if given an $H$-representation. To supplement these findings, we show that many classes of (distance) $(σ, ρ)$-problems are W[1]-hard parameterized by mim-width + solution size.

We give a first polynomial-time algorithm for (Weighted) Feedback Vertex Set on graphs of bounded maximum induced matching width (mim-width). Explicitly, given a branch decomposition of mim-width w, we give an n^{O(w)}-time algorithm that solves Feedback Vertex Set. This provides a unified algorithm for many well-known classes, such as Interval graphs and Permutation graphs, and furthermore, it gives the first polynomial-time algorithms for other classes of bounded mim-width, such as Circular Permutation and Circular k-Trapezoid graphs for fixed k. In all these classes the decomposition is computable in polynomial time, as shown by Belmonte and Vatshelle [Theor. Comput. Sci. 2013]. We show that powers of graphs of tree-width w-1 or path-width w and powers of graphs of clique-width w have mim-width at most w. These results extensively provide new classes of bounded mim-width. We prove a slight strengthening of the first statement which implies that, surprisingly, Leaf Power graphs which are of importance in the field of phylogenetic studies have mim-width at most 1. Given a tree decomposition of width w-1, a path decomposition of width w, or a clique-width w-expression of a graph G, one can for any value of k find a mim-width decomposition of its k-power in polynomial time, and apply our algorithm to solve Feedback Vertex Set on the k-power in time n^{O(w)}. In contrast to Feedback Vertex Set, we show that Hamiltonian Cycle is NP-complete even on graphs of linear mim-width 1, which further hints at the expressive power of the mim-width parameter.

We give the first polynomial-time algorithms on graphs of bounded maximum induced matching width (mim-width) for problems that are not locally checkable. In particular, we give $n^{\mathcal{O}(w)}$-time algorithms on graphs of mim-width at most $w$, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and $H$-Induced Topological Minor for fixed $H$. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Permutation and Circular Permutation graphs, Convex graphs, $k$-Trapezoid, Circular $k$-Trapezoid, $k$-Polygon, Dilworth-$k$ and Co-$k$-Degenerate graphs for fixed $k$.

One of the most famous algorithmic meta-theorems states that every graph property that can be defined by a sentence in counting monadic second order logic (CMSOL) can be checked in linear time for graphs of bounded treewidth, which is known as Courcelle's Theorem. These algorithms are constructed as finite state tree automata, and hence every CMSOL-definable graph property is recognizable. Courcelle also conjectured that the converse holds, i.e., every recognizable graph property is definable in CMSOL for graphs of bounded treewidth. We prove this conjecture for k-outerplanar graphs, which are known to have treewidth at most 3k-1.