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For a graph $G$, a subset $S\subseteq V(G)$ is called a resolving set of $G$ if, for any two vertices $u,v\in V(G)$, there exists a vertex $w\in S$ such that $d(w,u)\neq d(w,v)$. 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 $S\subseteq V(G)$ of size at most $k$ such that, for any vertex $u \in V(G)$, there are two vertices $s_1, s_2 \in S$ such that $u$ lies on a shortest path from $s_1$ to $s_2$. These two classical problems turn out 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. Some of the very few existing tractable results state that they are both FPT with respect to the vertex cover number $vc$. More precisely, we observe that both problems admit an FPT algorithm running in time $2^{\mathcal{O}(vc^2)}\cdot n^{\mathcal{O}(1)}$, and a kernelization algorithm that outputs a kernel with $2^{\mathcal{O}(vc)}$ vertices. We prove that unless the Exponential Time Hypothesis fails, Metric Dimension and Geodetic Set, even on graphs of bounded diameter, neither admit an FPT algorithm running in time $2^{o(vc^2)}\cdot n^{\mathcal(1)}$, nor a kernelization algorithm that reduces the solution size and outputs a kernel with $2^{o(vc)}$ vertices. The versatility of our technique enables us to apply it to both these problems. We only know of one other problem in the literature that admits such a tight lower bound. Similarly, the list of known problems with exponential lower bounds on the number of vertices in kernelized instances is very short., Comment: This is the second part following an accompanying paper arXiv:2307.08149. We split the original paper to keep paper length more manageable

We investigate a relaxation of the notion of fractional treewidth-fragility, namely fractional tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for meta-problems such as finding a maximum-weight sparse induced subgraph satisfying a given $\mathsf{CMSO}_2$ formula 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, and its applications to exact subexponential-time algorithms., Comment: 48 pages, 8 figures. arXiv admin note: substantial text overlap with arXiv:2303.07444

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, $\Sigma_2^p$, $\Pi_2^p$, 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., Comment: Shortened abstract to meet arxiv requirements. We split the original paper to keep paper length more manageable. This is the first part following an accompanying paper arXiv:2405.01344; and will be presented at ICALP 2024

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., Comment: Full version of a paper to appear in a shorter form in the 39th International Symposium on Computational Geometry (SoCG 2023)

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})}., Comment: 87 pages, 18 figures

The leafage of a chordal graph G is the minimum integer l such that G can be realized as an intersection graph of subtrees of a tree with 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 $2^{O(l^2)} n^{O(1)}$. We present a conceptually much simpler algorithm that runs in time $2^{O(l)} n^{O(1)}$. 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 W[1]-hard when parameterized by the leafage and complement this result by presenting a simple $n^{O(l)}$-time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in $n^{O(1)}$-time., Comment: 48 pages, 3 figures, 1 table

For a graph $G$, a subset $S \subseteq V(G)$ is called a \emph{resolving set} if for any two vertices $u,v \in V(G)$, there exists a vertex $w \in S$ such that $d(w,u) \neq d(w,v)$. The {\sc 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 \emph{smaller} parameters, in particular the feedback vertex set number. We answer this question by proving that {\sc 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 {\sc 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.

We consider the problem of reducing the (semi)total domination number of graph by one by contracting edges. It is known that this can always be done with at most three edge contractions and that deciding whether one edge contraction suffices is an $\mathsf{NP}$-hard problem. We show that for every fixed $k \in \{2,3\}$, deciding whether exactly $k$ edge contractions are necessary is $\mathsf{NP}$-hard and further provide for $k=2$ complete complexity dichotomies on monogenic graph classes., Comment: 38 pages, 13 figures

The Edge Multicut problem is a classical cut problem where given an undirected graph $G$, a set of pairs of vertices $\mathcal{P}$, and a budget $k$, the goal is to determine if there is a set $S$ of at most $k$ edges such that for each $(s,t) \in \mathcal{P}$, $G-S$ has no path from $s$ to $t$. Edge Multicut has been relatively recently shown to be fixed-parameter tractable (FPT), parameterized by $k$, by Marx and Razgon [SICOMP 2014], and independently by Bousquet et al. [SICOMP 2018]. In the weighted version of the problem, called Weighted Edge Multicut one is additionally given a weight function $\mathtt{wt} : E(G) \to \mathbb{N}$ and a weight bound $w$, and the goal is to determine if there is a solution of size at most $k$ and weight at most $w$. Both the FPT algorithms for Edge Multicut by Marx et al. and Bousquet et al. fail to generalize to the weighted setting. In fact, the weighted problem is non-trivial even on trees and determining whether Weighted Edge Multicut on trees is FPT was explicitly posed as an open problem by Bousquet et al. [STACS 2009]. In this article, we answer this question positively by designing an algorithm which uses a very recent result by Kim et al. [STOC 2022] about directed flow augmentation as subroutine. We also study a variant of this problem where there is no bound on the size of the solution, but the parameter is a structural property of the input, for example, the number of leaves of the tree. We strengthen our results by stating them for the more general vertex deletion version., Comment: Full version of the paper accepted for WG 2022

In this paper, we consider the problem of reducing the semitotal domination number of a given graph by contracting $k$ edges, for some fixed $k \geq 1$. 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.

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 $P_3+kP_2$-free graphs for any $k \geq 0$ which combined with results of [1,2] leads to 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($\gamma_t$)). 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 dichotomy for $H$-free graphs.

We consider Independent Set and Dominating Set restricted to VPG graphs (or, equivalently, string graphs). We show that they both remain $\mathsf{NP}$-hard on $B_0$-VPG graphs admitting a representation such that each grid-edge belongs to at most one path and each horizontal path has length at most two. On the other hand, combining the well-known Baker's shifting technique with bounded mim-width arguments, we provide simple PTASes on VPG graphs admitting a representation such that each grid-edge belongs to at most $t$ paths and the length of the horizontal part of each path is at most $c$, for some $c \geq 1$.

A contact $B_0$-VPG graph is a graph for which there exists a collection of nontrivial pairwise interiorly disjoint horizontal and vertical segments in one-to-one correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding segments touch. It was shown by Deniz et al. that Recognition is $\mathsf{NP}$-complete for contact $B_0$-VPG graphs. In this paper we present a minimal forbidden induced subgraph characterisation of contact $B_0$-VPG graphs within the class of circular-arc graphs and provide a polynomial-time algorithm for recognising these graphs.

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 $\mathsf{NP}$-hard when restricted to $\{P_6,P_4+P_2\}$-free graphs and that it is $\mathsf{coNP}$-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., Comment: arXiv admin note: text overlap with arXiv:1903.01800

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.

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 $k \geq 0$, the graph is said to be $B_k$-CPG. We first show that, for any $k \geq 0$, the class of $B_k$-CPG graphs is strictly contained in the class of $B_{k+1}$-CPG graphs even within the class of planar graphs, thus implying that there exists no $k \geq 0$ such that every planar CPG graph is $B_k$-CPG. The main result of the paper is that recognizing CPG graphs and $B_k$-CPG graphs with $k \geq 1$ is $\mathsf{NP}$-complete. Moreover, we show that the same remains true even within the class of planar graphs in the case $k \geq 3$. We then consider several graph problems restricted to CPG graphs and show, in particular, that Independent Set and Clique Cover remain $\mathsf{NP}$-hard for $B_0$-CPG graphs. Finally, we consider the related classes $B_k$-EPG of edge-intersection graphs of paths with at most $k$ bends on a grid. Although it is possible to optimally color a $B_0$-EPG graph in polynomial time, as this class coincides with that of interval graphs, we show that, in contrast, 3-Colorability is $\mathsf{NP}$-complete for $B_1$-EPG graphs.

For every fixed integer $k \geq 1$, we prove that $k$-Edge Colouring is fixed-parameter-tractable when parameterized by the number of vertices of maximum degree.

We show that for any $k \geq 0$, there exists a planar graph which is $B_{k+1}$-CPG but not $B_k$-CPG. As a consequence, we obtain that $B_k$-CPG is a strict subclass of $B_{k+1}$-CPG.

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 $\gamma_{t2}(G)$ of a semitotal dominating set of $G$ is squeezed between the domination number $\gamma(G)$ and the total domination number $\gamma_{t}(G)$. \textsc{Semitotal Dominating Set} is the problem of finding, given a graph $G$, a semitotal dominating set of $G$ of size $\gamma_{t2}(G)$. 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 with bounded mim-width by a reduction to \textsc{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 \textsc{Semitotal Dominating Set} and \textsc{Total Dominating Set}. Finally, we show that it is $\mathsf{NP}$-complete to recognise the graphs such that $\gamma_{t2}(G) = \gamma_{t}(G)$ and those such that $\gamma(G) = \gamma_{t2}(G)$, even if restricted to be planar and with maximum degree at most $4$, and we provide forbidden induced subgraph characterisations for the graphs heriditarily satisfying either of these two equalities.

A graph is $H$-free if it does not contain an induced subgraph isomorphic to $H$. For every integer $k$ and every graph $H$, we determine the computational complexity of $k$-Edge Colouring for $H$-free graphs.

In this paper we present a characterisation, by an infinite family of minimal forbidden induced subgraphs, of proper circular arc graphs which are intersection graphs of paths on a grid, where each path has at most one bend (turn).

In this paper we consider Contact graphs of Paths on a Grid (CPG graphs), i.e. graphs for which there exists a family of interiorly disjoint paths on a grid in one-to-one correspondence with their vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. Our class generalizes the well studied class of VCPG graphs (see [1]). We examine CPG graphs from a structural point of view which leads to constant upper bounds on the clique number and the chromatic number. Moreover, we investigate the recognition and 3-colorability problems for $B_0$-CPG, a subclass of CPG. We further show that CPG graphs are not necessarily planar and not all planar graphs are CPG., Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018)