ALCOMFT-TR-02-111

Naomi Nishimura, Prabhakar Ragde and Dimitrios M. Thilikos
Fast fixed-parameter tractable algorithms for nontrivial generalizations of vertex cover
Barcelona. Work package 2. May 2002.

Abstract: 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 \withinkG, 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 minor-closed classes for which all obstructions are connected graphs, and showing that the size of any obstruction for W_k(G) is O(tk⁷+t⁷k²), 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.

Postscript file: ALCOMFT-TR-02-111.ps.gz (89 kb).