Definability Equals Recognizability for kk-Outerplanar Graphs

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 $3k-1$.Comment: 40 pages, 8 figure

On the maximum number of edges in planar graphs of bounded degree and matching number

We determine the maximum number of edges that a planar graph can have as a function of its maximum degree and matching number.publishedVersio

MSOL-Definability Equals Recognizability for Halin Graphs and Bounded Degree k-Outerplanar Graphs

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 a number of special cases in a stronger form. That is, we show that each recognizable property is definable in MSOL, i.e. the counting operation is not needed in our expressions. We give proofs for Halin graphs, bounded degree k-outerplanar graphs and some related graph classes. We furthermore show that the conjecture holds for any graph class that admits tree decompositions that can be defined in MSOL, thus providing a useful tool for future proofs

Fine-grained parameterized complexity analysis of graph coloring problems

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 ∗(θ k) 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.</p

Fine-grained parameterized complexity analysis of graph coloring problems

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 ∗(θ k) 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.</p

On the Hardness of Generalized Domination Problems Parameterized by Mim-Width

For nonempty ?, ? ? ?, a vertex set S in a graph G is a (?, ?)-dominating set if for all v ? S, |N(v) ? S| ? ?, and for all v ? V(G) ? S, |N(v) ? S| ? ?. The Min/Max (?,?)-Dominating Set problems ask, given a graph G and an integer k, whether G contains a (?, ?)-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 ? and ?, the Min/Max (?,?)-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 ? and ? 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

bb-Coloring Parameterized by Pathwidth is XNLP-complete

We show that the $b$-Coloring problem is complete for the class XNLP 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

Structural Parameterizations of Clique Coloring

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 ? 2, we give an ?^?(q^{tw})-time algorithm when the input graph is given together with one of its tree decompositions of width 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 XP parameterized by clique-width

Generalized Distance Domination Problems and Their Complexity on Graphs of Bounded mim-width

We generalize the family of (sigma, rho)-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) (sigma, rho)-problems are W[1]-hard parameterized by mim-width + solution size