Abstract

A graph is well-indumatched if all its maximal induced matchings are of the same size. We first prove that recognizing whether a graph is well-indumatched is a co-NP-complete problem even for (2P5,K1,5)-free graphs. We then show that decision problems Independent Dominating Set, Independent Set, and Dominating Set are NP-complete for the class of well-indumatched graphs. We also show that this class is a co-indumatching hereditary class, i.e., it is closed under deleting the end-vertices of an induced matching along with their neighborhoods, and we characterize well-indumatched graphs in terms of forbidden co-indumatching subgraphs. We prove that recognizing a co-indumatching subgraph is an NP-complete problem. We introduce a perfectly well-indumatched graph, in which every induced subgraph is well-indumatched, and characterize the class of these graphs in terms of forbidden induced subgraphs. Finally, we show that the weighted versions of problems Independent Dominating Set and Independent Set can be solved in polynomial time for perfectly well-indumatched graphs, but problem Dominating Set is NP-complete.

Highlights

  • In this paper, we use graph-theoretic terminology of Bondy and Murty [7], and computational complexity terminology of Garey and Johnson [29].A matching in a graph is a set of edges with no two edges having a common vertex

  • We show that this class is a co-indumatching hereditary class, i.e., it is closed under deleting the end-vertices of an induced matching along with their neighborhoods, and we characterize well-indumatched graphs in terms of forbidden co-indumatching subgraphs

  • We show that the weighted versions of problems Independent Dominating Set and Independent Set can be solved in polynomial time for perfectly well-indumatched graphs, but problem Dominating Set is NP-complete

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Summary

Introduction

We use graph-theoretic terminology of Bondy and Murty [7] (unless noted otherwise), and computational complexity terminology of Garey and Johnson [29]. The problem of recognizing well-covered graphs is proved to be co-NP-complete for general graphs, independently by Sankaranarayana and Stewart [46], and Chvátal and Slater [20]. We prove that, for the same graphs, the problem of recognizing a graph that has maximal induced matchings of at most t distinct sizes is co-NP-complete for any given t 1. We characterize the class of perfectly well-indumatched graphs by a finite set of forbidden induced subgraphs, obtaining a new polynomial-time recognizable class, where both parameters σ and Σ are easy to compute.

Complexity of recognizing well-indumatched graphs
NP-completeness results for well-indumatched graphs
Co-indumatching subgraphs
Perfectly well-indumatched graphs
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