We show that, for every fixed positive integers $r$ and $k$, \textsc{Max-Weight List $r$-Colorable Induced Subgraph} admits a polynomial-time algorithm on $kP_3$-free graphs. This problem is a common generalization of \textsc{Max-Weight Independent Set}, \textsc{Odd Cycle Transversal} and \textsc{List $r$-Coloring}, among others. Our result has several consequences. First, it implies that, for every fixed $r \geq 5$, assuming $\mathsf{P}\neq \mathsf{NP}$, \textsc{Max-Weight List $r$-Colorable Induced Subgraph} is polynomial-time solvable on $H$-free graphs if and only if $H$ is an induced subgraph of either $kP_3$ or $P_5+kP_1$, for some $k \geq 1$. Second, it makes considerable progress toward a complexity dichotomy for \textsc{Odd Cycle Transversal} on $H$-free graphs, allowing to answer a question of Agrawal, Lima, Lokshtanov, Rz{ą}{ż}ewski, Saurabh, and Sharma [TALG 2024]. Third, it gives a short and self-contained proof of the known result of Chudnovsky, Hajebi, and Spirkl [Combinatorica 2024] that \textsc{List $r$-Coloring} on $kP_3$-free graphs is polynomial-time solvable for every fixed $r$ and $k$. We also consider two natural distance-$d$ generalizations of \textsc{Max-Weight Independent Set} and \textsc{List $r$-Coloring} and provide polynomial-time algorithms on $kP_3$-free graphs for every fixed integers $r$, $k$, and $d \geq 6$.

For a graph G, a subset S ⊆ V(G) is called a resolving set of G if, for any two vertices u,v ∈ V(G), there exists a vertex w ∈ S such that d(w,u) ≠ d(w,v). The Metric Dimension problem takes as input a graph G on n vertices and a positive integer k, and asks whether there exists a resolving set of size at most k. In another metric-based graph problem, Geodetic Set, the input is a graph G and an integer k, and the objective is to determine whether there exists a subset S ⊆ V(G) of size at most k such that, for any vertex u ∈ V(G), there are two vertices s₁, s₂ ∈ S such that u lies on a shortest path from s₁ to s₂. These two classical problems are known to be intractable with respect to the natural parameter, i.e., the solution size, as well as most structural parameters, including the feedback vertex set number and pathwidth. We observe that both problems admit an FPT algorithm running in 2^𝒪(vc²) ⋅ n^𝒪(1) time, and a kernelization algorithm that outputs a kernel with 2^𝒪(vc) vertices, where vc is the vertex cover number. We prove that unless the Exponential Time Hypothesis (ETH) fails, Metric Dimension and Geodetic Set, even on graphs of bounded diameter, do not admit - an FPT algorithm running in 2^o(vc²) ⋅ n^𝒪(1) time, nor - a kernelization algorithm that does not increase the solution size and outputs a kernel with 2^o(vc) vertices. We only know of one other problem in the literature that admits such a tight algorithmic lower bound with respect to vc. Similarly, the list of known problems with exponential lower bounds on the number of vertices in kernelized instances is very short.

In this paper, we present new efficiently solvable cases of the Minimum Uncovering Branching problem, an optimization problem with applications in cancer genomics introduced by Hujdurović, Husić, Milanič, Rizzi, and Tomescu in 2018. The problem involves a family of finite sets, and the goal is to map each non-maximal set to exactly one set that contains it, minimizing the sum of uncovered elements across all sets in the family. Hujdurović et al. formulated the problem in terms of branchings of the digraph formed by the proper set inclusion relation on the input sets and studied the problem complexity based on properties of the corresponding partially ordered set, in particular, with respect to its height and width, defined respectively as the maximum cardinality of a chain and an antichain. They showed that the problem is APX-complete for instances of bounded height and that a constant-factor approximation algorithm exists for instances of bounded width, but left the exact complexity for bounded-width instances open. In this paper, we answer this question by proving that the problem is solvable in polynomial time. We derive this result by examining the structural properties of optimal solutions and reducing the problem to computing maximum matchings in bipartite graphs and maximum weight antichains in partially ordered sets. We also introduce a new polynomially computable lower bound and identify another condition for polynomial-time solvability.

Treewidth (tw) is an important parameter that, when bounded, yields tractability for many problems. For example, graph problems expressible in Monadic Second Order (MSO) logic and QUANTIFIED SAT or, more generally, QUANTIFIED CSP, are FPT parameterized by the tw of the input's (primal) graph plus the length of the MSO-formula [Courcelle, Information & Computation 1990] and the quantifier rank [Chen, ECAI 2004], resp. The algorithms from these (meta-)results have running times whose dependence on tw is a tower of exponents. A conditional lower bound by Fichte et al. [LICS 2020] shows that, for QUANTIFIED SAT, the height of this tower is equal to the number of quantifier alternations. Lower bounds showing that at least double-exponential factors in the running time are necessary are rare: there are very few (for tw and vertex cover vc parameterizations) and they are for problems that are complete for #NP, $Σ_2^p$, $Π_2^p$, or higher levels of the polynomial hierarchy. We show, for the first time, that it is not necessary to go higher up in the polynomial hierarchy to obtain such lower bounds. We design a novel, yet simple versatile technique based on Sperner families to obtain such lower bounds and apply it to 3 problems: METRIC DIMENSION, STRONG METRIC DIMENSION, and GEODETIC SET. We prove that they do not admit $2^{2^{o(tw)}} \cdot n^{O(1)}$-time algorithms, even on bounded diameter graphs, unless the ETH fails. For STRONG METRIC DIMENSION, the lower bound holds even for vc. We complement our lower bounds with matching upper bounds.

In the Directed Steiner Network problem, the input is a directed graph G, a subset T of k vertices of G called the terminals, and a demand graph D on T. The task is to find a subgraph H of G with the minimum number of edges such that for every edge (s,t) in D, the solution H contains a directed s to t path. In this paper we investigate how the complexity of the problem depends on the demand pattern when G is planar. Formally, if \mathcal{D} is a class of directed graphs closed under identification of vertices, then the \mathcal{D}-Steiner Network (\mathcal{D}-SN) problem is the special case where the demand graph D is restricted to be from \mathcal{D}. For general graphs, Feldmann and Marx [ICALP 2016] characterized those families of demand graphs where the problem is fixed-parameter tractable (FPT) parameterized by the number k of terminals. They showed that if \mathcal{D} is a superset of one of the five hard families, then \mathcal{D}-SN is W[1]-hard parameterized by k, otherwise it can be solved in time f(k)n^{O(1)}. For planar graphs an interesting question is whether the W[1]-hard cases can be solved by subexponential parameterized algorithms. Chitnis et al. [SICOMP 2020] showed that, assuming the ETH, there is no f(k)n^{o(k)} time algorithm for the general \mathcal{D}-SN problem on planar graphs, but the special case called Strongly Connected Steiner Subgraph can be solved in time f(k) n^{O(\sqrt{k})} on planar graphs. We present a far-reaching generalization and unification of these two results: we give a complete characterization of the behavior of every $\mathcal{D}$-SN problem on planar graphs. We show that assuming ETH, either the problem is (1) solvable in time 2^{O(k)}n^{O(1)}, and not in time 2^{o(k)}n^{O(1)}, or (2) solvable in time f(k)n^{O(\sqrt{k})}, but not in time f(k)n^{o(\sqrt{k})}, or (3) solvable in time f(k)n^{O(k)}, but not in time f(k)n^{o({k})}.

Abstract The leafage of a chordal graph G is the minimum integer $$\ell $$ ℓ such that G can be realized as an intersection graph of subtrees of a tree with $$\ell $$ ℓ leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time $$2^{\mathcal {O}(\ell ^2)} \cdot n^{\mathcal {O}(1)}$$ 2 O ( ℓ 2 ) · n O ( 1 ) . We present a conceptually much simpler algorithm that runs in time $$2^{\mathcal {O}(\ell )} \cdot n^{\mathcal {O}(1)}$$ 2 O ( ℓ ) · n O ( 1 ) . We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is [1]-hard when parameterized by the leafage and complement this result by presenting a simple $$n^{\mathcal {O}(\ell )}$$ n O ( ℓ ) -time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in $$n^{{{\mathcal {O}}}(1)}$$ n O ( 1 ) -time.

We investigate a relaxation of the notion of treewidth-fragility, namely tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for independent packing problems on fractionally tree-independence-number-fragile graph classes. Our approach unifies and extends several known polynomial-time approximation schemes on seemingly unrelated graph classes, such as classes of intersection graphs of fat objects in a fixed dimension or proper minor-closed classes. We also study the related notion of layered tree-independence number, a relaxation of layered treewidth.

Abstract. For a graph [Formula: see text], a subset [Formula: see text] is called a resolving set if for any two vertices [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text]. The Metric Dimension problem takes as input a graph [Formula: see text] and a positive integer [Formula: see text], and asks whether there exists a resolving set of size at most [Formula: see text]. This problem was introduced in the 1970s and is known to be NP -hard [M. R. Garey and D. S. Johnson, Computers and Intractability—A Guide to NP-Completeness, Freeman, San Francisco, 1979]. In the realm of parameterized complexity, Hartung and Nichterlein [28 th Conference on Computational Complexity, IEEE, Piscataway, NJ, 2013, pp. 266–276] proved that the problem is W [2]-hard when parameterized by the natural parameter [Formula: see text]. They also observed that it is fixed parameter tractable ( FPT) when parameterized by the vertex cover number and asked about its complexity under smaller parameters, in particular, the feedback vertex set number. We answer this question by proving that Metric Dimension is W [1]-hard when parameterized by the combined parameter feedback vertex set number plus pathwidth. This also improves the result of Bonnet and Purohit [IPEC 2019] which states that the problem is W [1]-hard parameterized by the pathwidth. On the positive side, we show that Metric Dimension is FPT when parameterized by either the distance to cluster or the distance to cocluster, both of which are smaller parameters than the vertex cover number.

In this paper, we consider the problem of reducing the semitotal domination number of a given graph by contracting k edges, for some fixed k≥1. We show that this can always be done with at most 3 edge contractions and further characterise those graphs requiring 1, 2 or 3 edge contractions, respectively, to decrease their semitotal domination number. We then study the complexity of the problem for k=1 and obtain in particular a complete complexity dichotomy for monogenic classes.

For a graph G, a subset S ⊆ V(G) is called a resolving set if for any two vertices u,v ∈ V(G), there exists a vertex w ∈ S such that d(w,u) ≠ d(w,v). The Metric Dimension problem takes as input a graph G and a positive integer k, and asks whether there exists a resolving set of size at most k. This problem was introduced in the 1970s and is known to be NP-hard [GT 61 in Garey and Johnson’s book]. In the realm of parameterized complexity, Hartung and Nichterlein [CCC 2013] proved that the problem is W[2]-hard when parameterized by the natural parameter k. They also observed that it is FPT when parameterized by the vertex cover number and asked about its complexity under smaller parameters, in particular the feedback vertex set number. We answer this question by proving that Metric Dimension is W[1]-hard when parameterized by the feedback vertex set number. This also improves the result of Bonnet and Purohit [IPEC 2019] which states that the problem is W[1]-hard parameterized by the treewidth. Regarding the parameterization by the vertex cover number, we prove that Metric Dimension does not admit a polynomial kernel under this parameterization unless NP ⊆ coNP/poly. We observe that a similar result holds when the parameter is the distance to clique. On the positive side, we show that Metric Dimension is FPT when parameterized by either the distance to cluster or the distance to co-cluster, both of which are smaller parameters than the vertex cover number.

In this paper we continue the systematic study of Contact graphs of Paths on a Grid (CPG graphs) initiated in Deniz et al. (2018). A CPG graph is a graph for which there exists a collection of pairwise interiorly disjoint paths on a grid in one-to-one correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. If every such path has at most k bends for some k≥0, the graph is said to be Bk-CPG. We first show that, for any k≥0, the class of Bk-CPG graphs is strictly contained in the class of Bk+1-CPG graphs even within the class of planar graphs, thus implying that there exists no k≥0 such that every planar CPG graph is Bk-CPG. The main result of the paper is that recognizing CPG graphs and Bk-CPG graphs with k≥1 is NP-complete. Moreover, we show that the same remains true even within the class of planar graphs in the case k≥3. We then consider several graph problems restricted to CPG graphs and show, in particular, that Independent Set and Clique Cover remain NP-hard for B0-CPG graphs. Finally, we consider the related classes Bk-EPG of edge-intersection graphs of paths with at most k bends on a grid. Although it is possible to optimally color a B0-EPG graph in polynomial time, as this class coincides with that of interval graphs, we show that, in contrast, 3-Colorability is NP-complete for B1-EPG graphs.

In this note, we consider the following problem: given a connected graph G, can we reduce the domination number of G by using only one edge contraction? We show that the problem is polynomial-time solvable on (P3+kP2)-free graphs for any k≥0 which can be combined with former results to obtain a complexity dichotomy of the problem on H-free graphs.

In this paper, we study the problem of deciding whether the total domination number of a given graph G can be reduced using exactly one edge contraction (called 1-Edge Contraction(γt)). We focus on several graph classes and determine the computational complexity of this problem. By putting together these results, we manage to obtain a complete complexity dichotomy for H-free graphs.

A semitotal dominating set of a graph G with no isolated vertex is a dominating set D of G such that every vertex in D is within distance two of another vertex in D. The minimum size γt2(G) of a semitotal dominating set of G is squeezed between the domination number γ(G) and the total domination number γt(G). Semitotal Dominating Set is the problem of finding, given a graph G, a semitotal dominating set of G of size γt2(G). In this paper, we continue the systematic study on the computational complexity of this problem when restricted to special graph classes. In particular, we show that it is solvable in polynomial time for the class of graphs of bounded mim-width by a reduction to Total Dominating Set and we provide several approximation lower bounds for subclasses of subcubic graphs. Moreover, we obtain complexity dichotomies in monogenic classes for the decision versions of Semitotal Dominating Set and Total Dominating Set. Finally, we show that it is NP-complete to recognise the graphs such that γt2(G)=γt(G) and those such that γ(G)=γt2(G), even if restricted to be planar and with maximum degree at most 4, and we provide forbidden induced subgraph characterisations for the graphs hereditarily satisfying either of these two equalities.

In this paper, we consider the following problem: given a connected graph G, can we reduce the domination number of G by one by using only one edge contraction? We show that the problem is NP-hard when restricted to {P_6,P_4+P_2}-free graphs and that it is coNP-hard when restricted to subcubic claw-free graphs and 2P_3-free graphs. As a consequence, we are able to establish a complexity dichotomy for the problem on H-free graphs when H is connected.

In this paper, we study the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by at least one using $k$ edge contractions, for some fixed integer $k \geq 0$? We present positive and negative results regarding the computational complexity of this problem.

We investigate the Matrix Powering Positivity Problem, PosMatPow: given an m X m square integer matrix M, a linear function f: Z^{m X m} -> Z with integer coefficients, and a positive integer n (encoded in binary), determine whether f(M^n) \geq 0. We show that for fixed dimensions m of 2 and 3, this problem is decidable in polynomial time.