Graphs with Maximal Induced Matchings of the Same Size

Abstract

Abstract. A graph is well-indumatched if all its maximal (with respect to set inclusion) induced matchings are of the same size. We first prove that recognizing the class WIM of well-indumatched graphs is a co-NP-complete problem even for (2P5, K1,5)-free graphs. We then show that the well-known decision problems such as Independent Dominating Set, Independent Set, and Dominating Set are NP-complete for well-indumatched graphs. We also show that WIM is a co-indumatching hereditary class and characterize well-indumatched graphs in terms of forbidden co-indumatching subgraphs. However, we prove that recognizing co-indumatching subgraphs is an NP-complete problem. A graph G is perfectly well-indumatched if every induced subgraph of G is well-indumatched. We characterize the class of perfectly well-indumatched graphs in terms of forbidden induced subgraphs. Finally, we show that both Independent Dominating Set and Independent Set can be solved in polynomial time for perfectly well-indumatched graphs, even in their weighted versions, but Dominating Set is still NP-complete.

On Minimum Maximal Distance-k Matchings

“…The paper is organized as follows. In Section 2 we show that the recognition of k-equimatchable graphs is co-NP-complete, generalizing the result of Baptiste et al (2017) from k = 2 to the case of an arbitrary k ≥ 2. It also follows from the proof that the problem of finding a minimum maximal distance-k matching in a graph is NP-hard for every fixed k ≥ 2.…”

“…In this section we prove that the recognition of k-equimatchable graphs is a co-NP-complete problem for any fixed k ≥ 2 by constructing a polynomial time reduction from an NP-hard 3SAT problem [Garey and Johnson (1990)]. We generalize the reduction proposed by Baptiste et al (2017) for 2-equimatchable graphs, which is in turn a generalization of the technique proposed by Chvátal and Slater (1993).…”

“…See Plummer (1993) for a survey of further related results. In Baptiste et al (2017) it is proved that the recognition of 2-equimatchable graphs is a co-NP-complete problem.…”

Well-indumatched Trees and Graphs of Bounded Girth

“…An induced matching is a matching M such that no two edges of M is joined by an edge, in other words, M occurs as an induced subgraph of G. In this paper, we are interested in graphs such that all their inclusion-wise maximal induced matchings have the same size. These graphs have been introduced very recently in [2], where they are called well-indumatched graphs.…”

“…This problem is called Minimum Maximal Induced Matching and denoted by MMIM for short. MMIM has been shown to be NP-hard even in bipartite graphs of maximum degree 4 [27] or in graphs having all of their maximal induced matchings of size either k or k +1 for some integer k ≥ 1 [2]. The generalization of MMIM to distance-k matchings has been also considered recently [22].…”