Critical vertices and edges in H-free graphs

Abstract

A vertex or edge in a graph is critical if its deletion reduces the chromatic number of the graph by 1. We consider the problems of deciding whether a graph has a critical vertex or edge, respectively. We give a complexity dichotomy for both problems restricted to H-free graphs, that is, graphs with no induced subgraph isomorphic to H. Moreover, we show that an edge is critical if and only if its contraction reduces the chromatic number by 1. Hence, we also obtain a complexity dichotomy for the problem of deciding if a graph has an edge whose contraction reduces the chromatic number by 1.

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

“…Recall that Deletion Blocker(χ) and Contraction Blocker(χ) are called Critical Vertex and Contraction-Critical Edge, respectively, if d = k = 1. We need the following result announced in [36]; see [35] for its proof. We also need the following result of Král', Kratochvíl, Tuza, and Woeginger.…”

Blocking dominating sets for $H$-free graphs via edge contractions

“…In this paper, we continue the systematic study of the computational complexity of 1-Edge Contraction(γ) initiated in [9]. Ultimately, the aim is to obtain a complete classification for 1-Edge Contraction(γ) restricted to H-free graphs, for any (not necessarily connected) graph H, as it has been done for other blocker problems (see for instance [8,18,19]). As a step towards this end, we prove the following three theorems.…”

“…Such a designation follows from the fact that the set of vertices or edges involved can be viewed as "blocking" the parameter π. Identifying such sets may provide information on the structure of the input graph; for instance, if π = α, k = d = 1 and O = {vertex deletion}, the problem is equivalent to testing whether the input graph contains a vertex that is in every maximum independent set (see [18]). Blocker problems have received much attention in the recent literature (see for instance [1,2,3,4,5,7,8,9,11,12,13,15,16,17,18,19]) and have been related to other well-known graph problems such as Hadwiger Number, Club Contraction and several graph transversal problems (see for instance [7,17]). The graph parameters considered so far in the literature are the chromatic number, the independence number, the clique number, the matching number and the vertex cover number while the set O is a singleton consisting of a vertex deletion, edge contraction, edge deletion or edge addition.…”

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.…”