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

Abstract

Citation for published item:ulusmD hni¤ el nd iouleuD ghristophe nd iesD fernrd @PHIUA 9floking independent sets for rEfree grphs vi edge ontrtions nd vertex deletionsF9D in heory nd epplitions of wodels of gomputtionD IRth ennul gonferene @ewg PHIUA X fernD witzerlndD epril PHEPPD PHIU Y proeedingsF ghmX pringerD ppF RUHERVQF veture xotes in gomputer ieneF @IHIVSAF Further information on publisher's website:Publisher's copyright statement:The nal publication is available at Springer via https://doi.org/10.1007/978-3-319-55911-734Additional 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-prot 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. Let d and k be two given integers, and let G be a graph.Can we reduce the independence number of G by at least d via at most k graph operations from some fixed set S? This problem belongs to a class of so-called blocker problems. It is known to be co-NP-hard even if S consists of either an edge contraction or a vertex deletion. We further investigate its computational complexity under these two settings: -we give a sufficient condition on a graph class for the vertex deletion variant to be co-NP-hard even if d = k = 1; -in addition we prove that the vertex deletion variant is co-NP-hard for triangle-free graphs even if d = k = 1; -we prove that the edge contraction variant is NP-hard for bipartite graphs but linear-time solvable for trees. By combining our new results with known ones we are able to give full complexity classifications for both variants restricted to H-free graphs.

Reducing the Chromatic Number by Vertex or Edge Deletions

“…The complexities of Vertex Deletion Blocker(χ) and Contraction Blocker(χ) are known for H-free graphs if H is connected [10]. As a consequence of our results for k = d = 1, we can complete these two classifications for all graphs H, just as we did in a previous paper [11] for π = α (independence number) and π = ω (clique number), except for the case when π = ω, S = {ec} and H = C 3 + P 1 . The Edge Deletion Blocker(χ) problem is known [1] to be polynomial-time solvable for threshold graphs and NP-hard for cobipartite graphs.…”

“…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.…”

Critical vertices and edges in H-free graphs

“…Then, for a given graph G and integer k ≥ 0, the S-Blocker(π) problem asks if G can be modified into a graph G by using at most k operations from S so that π(G ) ≤ π(G) − d for some given threshold d ≥ 0. Over the last few years, the S-Blocker(π) problem has been well studied, see for instance [1,2,3,7,9,19,20,21,22,23]. If S consists of a single operation that is either a vertex deletion or edge contraction, then S-Blocker(π) is called Vertex Deletion Blocker(π) or Contraction Blocker(π), respectively.…”

The complexity of blocking (semi)total dominating sets with edge contractions

“…A blocker problem asks whether given a graph G, a graph a parameter π, a set O of one or more graph operations and an integer k ≥ 1, G can be transformed into a graph H such that π(H) ≤ π(G) − d for some threshold d ≥ 1, by using at most k operations from O. These problems are so called because the set of vertices or edges involved can be seen as "blocking" the parameter π. Identifying such sets may provide important information on the structure of the input graph: for instance, if π = α, k = d = 1 and O = {vertex deletion}, then the problem is equivalent to testing whether the input graph contains a vertex which belongs to every maximum independent set [20]. While the set O generally consists of a single operation (namely vertex deletion, edge deletion, edge addition or edge contraction), a variety of parameters have been considered in the literature including the chromatic number [2,5,6,19,21], the stability number [1,20], the clique number [16,17], the matching number [3,25], domination-like parameters [7,9,10,11,12,18] and others [4,13,14,22,24].…”