Reducing the Clique and Chromatic Number via Edge Contractions and Vertex Deletions

Abstract

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. We consider the following problem: can a certain graph parameter of some given graph G be reduced by at least d, for some integer d, via at most k graph operations from some specified set S, for some given integer k? As graph parameters we take the chromatic number and the clique number. We let the set S consist of either an edge contraction or a vertex deletion. As all these problems are NP-complete for general graphs even if d is fixed, we restrict the input graph G to some special graph class. We continue a line of research that considers these problems for subclasses of perfect graphs, but our main results are full classifications, from a computational complexity point of view, for graph classes characterized by forbidding a single induced connected subgraph H.

Blocking Independent Sets for H-Free Graphs via Edge Contractions and Vertex Deletions

“…We observe that Deletion Blocker(α) restricted to H-free graphs is equivalent to Deletion Blocker(ω) for H-free graphs, where H denotes the complement of H. Hence, as a corollary, we obtain an extension of the aforementioned classification of [14] for Deletion Blocker(ω) for H-free graphs from connected graphs H to all graphs H.…”

“…That is, we give a sufficient condition on such a graph class G, such that Deletion Blocker(α) is co-NP-hard for G even if d = k = 1. This condition is similar to a previous condition when π ∈ {χ, ω} [14]. In Section 6 we combine our new results from Sections 4 and 5 with known ones to obtain the classifications for H-free graphs.…”

“…We showed both new hardness results (e.g., for the class of perfect graphs itself) and tractable results (e.g., for cographs). In a follow-up paper [14] we extended the results of [7] by considering some more subclasses of perfect graphs for π ∈ {ω, χ}. Moreover, for every connected graph H and π ∈ {ω, χ}, we determined the computational complexity of Contraction Blocker(π) and Deletion Blocker(π) for H-free graphs, that is, graphs that do not contain an induced subgraph isomorphic to H.…”

“…In Section 6 we combine our new results from Sections 4 and 5 with known ones to obtain the classifications for H-free graphs. In Section 7 we compare our new results with the results of our previous paper [14] and list some open problems.…”

“…Blocker problems have been well studied in the recent literature [1-3, 5, 7, 14, 13, 16, 18]; in particular, in [7,14] several relations to other graph problems were identified, such as Hadwiger Number, Club Contraction and a number of graph transversal problems. So far, the graph parameters considered were the chromatic number, the independence number, the clique number, the matching number and the vertex cover number, whereas the set S consisted of a single graph operation, which was either the vertex deletion, edge contraction, edge deletion or the edge addition operation.…”