Contraction Blockers for Graphs with Forbidden Induced Paths

Abstract

We consider the following problem: can a certain graph parameter of some given graph be reduced by at least $$d$$ for some integeri¾ź$$d$$ via at most $$k$$ edge contractions for some given integer $$k$$? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when $$d$$ is part of the input, this problem is polynomial-time solvable on $$P_4$$-free graphs and NP-complete as well as W[1]-hard, with parameter $$d$$, for split graphs. As split graphs form a subclass of $$P_5$$-free graphs, both results together give a complete complexity classification for $$P_\ell $$-free graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameteri¾ź$$d$$. But we do show, on the positive side, that the problem is polynomial-time solvable, for each parameter, on split graphs if $$d$$ is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs.