Fast fixed-parameter tractable algorithms for nontrivial generalizations of vertex cover

• Authors: Naomi Nishimura;Prabhakar Ragde;Dimitrios M. Thilikos
• Venue: Discrete Applied Mathematics
• Year: 2005

Our goal in this paper is the development of fast algorithms for recognizing general classes of graphs. We seek algorithms whose complexity can be expressed as a linear function of the graph size plus an exponential function of k, a natural parameter describing the class. In particular, we consider the class W"k(G), where for each graph G in W"k(G), the removal of a set of at most k vertices from G results in a graph in the base graph class G. (If G is the class of edgeless graphs, W"k(G) is the class of graphs with bounded vertex cover.) When G is a minor-closed class such that each graph in G has bounded maximum degree, and all obstructions of G (minor-minimal graphs outside G) are connected, we obtain an O((g+k)|V(G)|+(fk)^k) recognition algorithm for W"k(G), where g and f are constants (modest and quantified) depending on the class G. If G is the class of graphs with maximum degree bounded by D (not closed under minors), we can still obtain a running time of O(|V(G)|(D+k)+k(D+k)^k^+^3) for recognition of graphs in W"k(G). Our results are obtained by considering bounded-degree minor-closed classes for which all obstructions are connected graphs, and showing that the size of any obstruction for W"k(G) is O(tk^7+t^7k^2), where t is a bound on the size of obstructions for G. A trivial corollary of this result is an upper bound of (k+1)(k+2) on the number of vertices in any obstruction of the class of graphs with vertex cover of size at most k. These results are of independent graph-theoretic interest.