VC Density of Set Systems Definable in Tree-Like Graphs

We study set systems definable in graphs using variants of logic with different expressive power. Our focus is on the notion of Vapnik-Chervonenkis density: the smallest possible degree of a polynomial bounding the cardinalities of restrictions of such set systems. On one hand, we prove that if phi(x,y) is a fixed CMSO_1 formula and C is a class of graphs with uniformly bounded cliquewidth, then the set systems defined by phi in graphs from C have VC density at most |y|, which is the smallest bound that one could expect. We also show an analogous statement for the case when phi(x,y) is a CMSO_2 formula and C is a class of graphs with uniformly bounded treewidth. We complement these results by showing that if C has unbounded cliquewidth (respectively, treewidth), then, under some mild technical assumptions on C, the set systems definable by CMSO_1 (respectively, CMSO_2) formulas in graphs from C may have unbounded VC dimension, hence also unbounded VC density

From tree-decompositions to clique-width terms

International audienceTree-width and clique-width are two important graph complexity measures that serve as parameters in many fixed-parameter tractable algorithms. We give two algorithms that transform tree-decompositions represented by normal trees into clique-width terms. As a consequence, we obtain that for sparse graphs, clique-width is polynomially bounded in terms of tree-width. It is even linearly bounded for planar graphs and incidence graphs. These results have applications to model-checking algorithms for problems described by monadic second-order formulas, including those allowing edge set quantifications

From tree-decompositions to clique-width terms

International audienceTree-width and clique-width are two important graph complexity measures that serve as parameters in many fixed-parameter tractable algorithms. We give two algorithms that transform tree-decompositions represented by normal trees into clique-width terms. As a consequence, we obtain that for sparse graphs, clique-width is polynomially bounded in terms of tree-width. It is even linearly bounded for planar graphs and incidence graphs. These results have applications to model-checking algorithms for problems described by monadic second-order formulas, including those allowing edge set quantifications