Abstract
In this work, we study the following problem: given a connected graph G, can we reduce the domination number of G by at least one using k edge contractions, for some fixed integer k>0? We show that for k=1 (resp. k=2), the problem is NP-hard (resp. coNP-hard). We further prove that for k=1, the problem is W[1]-hard parameterized by domination number plus the mim-width of the input graph, and that it remains NP-hard when restricted to chordal {P6,P4+P2}-free graphs, bipartite graphs and {C3,…,Cℓ}-free graphs for any ℓ≥3. We also show that for k=1, the problem is coNP-hard on subcubic claw-free graphs, subcubic planar graphs and on 2P3-free graphs. On the positive side, we show that for any k≥1, the problem is polynomial-time solvable on (P5+pK1)-free graphs for any p≥0 and that it can be solved in FPT-time and XP-time when parameterized by treewidth and mim-width, respectively. Finally, we start the study of the problem of reducing the domination number of a graph via vertex deletions and edge additions and, in this case, present a complexity dichotomy on H-free graphs.
Highlights
In a graph modification problem, we are usually interested in modifying a given graph G, via a small number of operations, into some other graph G′ that has a certain desired property
Such problems are called blocker problems as the set of vertices or edges involved can be viewed as ‘‘blocking’’ the parameter π. Identifying such sets may provide important information about 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
We show that even for k = 1, Vertex Deletion(γ ) is NP-hard and W[1]-hard parameterized by γ on split graphs, which rules out the possibility of algorithms running in FPT- or even XP-time parameterized by k for this problem, unless P=NP
Summary
In a graph modification problem, we are usually interested in modifying a given graph G, via a small number of operations, into some other graph G′ that has a certain desired property. While the set O consists of a single graph operation, namely either vertex deletion, edge contraction, edge deletion or edge addition Since these blocker problems are often NP-hard in general graphs, particular attention has been paid to their computational complexity when restricted to special graph classes. We study another parameter, namely the domination number γ , and we consider the following three operations: edge contraction, vertex deletion and edge addition, the first one being the main focus of our work. If k ≥ 3, k-Edge Contraction(γ ) has a simple answer: every graph with domination number at least two is a Yes-instance to this problem For this reason, we only consider the problem for k = 1, 2. We provide a few cases in which 1-Vertex Deletion(γ ) becomes polynomial-time solvable; in particular, we show that this is the case for (P4 + kP1)-free graphs.
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