Some New Results on Efficient and Perfect Edge Domination of Graphs

Abstract

A neighborhood star-free graph is one in which every vertex of degree at least two is contained in a triangle. This class has been considered before, in the context of edge domination. In the present paper, we study the complexity of the dominating induced matching problem for the class, and prove that the corresponding decision problem is NP-complete for several subclasses of neighborhood star-free graphs. In addition, we show that the dominating induced matching problem can be solved in polynomial time for neighborhood star-free graphs, containing no crickets as induced subgraphs.keyword: neighborhood-star-free graphs; triangle; efficient edge domination; perfect edge domination; dominating induced matching; crickets; cubic planar monotone 1in3SAT; planar C 4 -free positive 1in3SAT