Contraction Blockers for Graphs with Forbidden Induced Paths

Abstract

hinerD ¤ yF nd ulusm D hF nd i oule uD gF nd iesD fF @PHISA 9gontr tion lo kers for gr phs with for idden indu ed p thsF9D in elgorithms nd omplexity X Wth sntern tion l gonferen eD gseg PHISD risD pr n eD w y PHEPPD PHIS Y pro eedingsF D ppF IWREPHUF ve ture notes in omputer s ien eF @WHUWAF Further information on publisher's website:Publisher's copyright statement:The nal publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-18173-814.Additional information: Use policyThe 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 be reduced by at least d for some integer 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 P4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs. As split graphs form a subclass of P5-free graphs, both results together give a complete complexity classification for Pfree graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter d. But we do show, on the positive side, that the problem is polynomialtime 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.

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

“…In particular, Boros, Golumbic and Levit [4] proved that computing if the core of a graph has size at least is co-NP-hard for every fixed ≥ 1. Taking = 1 gives co-NP-hardness of 1-Deletion Blocker(α), whereas 1-Contraction Blocker(α) is known to be NP-hard [7]. Due to the above hardness results, it is natural to restrict the input to some special graph class.…”

“…Due to the above hardness results, it is natural to restrict the input to some special graph class. In a previous paper [7] we considered π ∈ {α, ω, χ}, where χ denotes the chromatic number of a graph, and we restricted the input to perfect graphs and subclasses of perfect graphs. We showed both new hardness results (e.g., for the class of perfect graphs itself) and tractable results (e.g., for cographs).…”

“…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.…”

“…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.…”

Reducing the Chromatic Number by Vertex or Edge Deletions

“…Such problems are called blocker problems, as the vertices or edges involved "block" some desirable graph property (such as being colorable with only a few colors). Over the last few years, blocker problems have been well studied, see for instance [1,2,3,4,5,9,10,11,12]. In these papers the set S consists of a single operation that is either a vertex deletion vd, edge deletion ed, or edge contraction ec.…”