Contraction and deletion blockers for perfect graphs and H-free graphs

Abstract

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Reducing graph transversals via edge contractions

“…When k and d are fixed instead of being part of the input, we denote the corresponding problem by k-Contraction(π, d). Blocker problems with the edge contraction operation have already been studied with respect to the chromatic number, clique number, and independence number [16,32], and the domination number [21,22], denoted by χ, ω, α, and γ, respectively. These works address the problem from the point of view of graph classes.…”

“…These works address the problem from the point of view of graph classes. Diner et al [16] showed, among other results, that Contraction(π) is NP-complete restricted 64:3 to split graphs for π ∈ {χ, α, ω}, but it is polynomial-time solvable in this graph class for fixed d in all three cases. Galby et al [21,22] recently initiated the study of the problem for π = γ for the case d = 1, providing several negative and positive results restricted to particular graph classes, such as a polynomial-time algorithm for k-Contraction(γ, 1) on (P 5 + pK 1 )-free graphs, for any p ≥ 1.…”

Assistance and Interdiction Problems on Interval Graphs

“…. , n and some large constant M , our problem is equivalent to the most vital nodes problem for independent set which was studied under the name independence number blocker problem by Diner et al [10]. They left the complexity of this problem as an open question.…”

“…Diner et al [10] proved that the most vital nodes problem for maximum clique size on interval graphs can be solved in O(n) time with a simple greedy algorithm [10]. Note that the most vital nodes problem for maximum clique size is a special case of the interdiction problem for maximum clique size in the shrink-expand framework by setting a i = b i for all i.…”

“…[1,2,4,7,17].) Diner et al [10] initiated the investigation of the most vital nodes problem on interval graphs. With respect to the clique number, they proved that a simple greedyalgorithm solves the most vital nodes problem on interval graphs.…”