Treewidth and Pathwidth Parameterized by the Vertex Cover Number

Abstract

International audienceAfter the number of vertices, Vertex Cover Number is the largest of the classical graph parameters and has more and more frequently been used as a separate parameter in parameterized problems, including problems that are not directly related to the Vertex Cover Number. Here we consider the treewidth and pathwidth problems parameterized by k, the size of a minimum vertex cover of the input graph. We show that the pathwidth and treewidth can be computed in O *(3 k ) time. This complements recent polynomial kernel results for treewidth and pathwidth parameterized by the Vertex Cover Number

Maximization Problems Parameterized Using Their Minimization Versions: The Case of Vertex Cover

“…Kernels for VC and WVC were given in [10,14,25], and results relating to the parameterized approximability of VC were also examined in the literature (see, e.g., [5,6,20]). Finally, in the context of Parameterized Complexity, we would also like to note that vc is a parameter of interest; indeed, apart from VC, there are other problems whose parameterized complexity was studied with respect to this parameter (see, e.g., [9,23,26]).…”

Algorithms Parameterized by Vertex Cover and Modular Width, Through Potential Maximal Cliques

“…Pipelined with our main combinatorial result, we deduce that all these problems can be solved in time O * (4 vc ) or O * (1.7347 mw ). Recently Chapelle et al [6] provided an algorithm solving Treewidth and Pathwidth in O * (3 vc ), but those completely different techniques do not seem to work for Minimum Fill-in or Treelength. The interested reader may also refer., e.g., to [10,11] for more (layout) problems parameterized by vertex cover.…”