"Structure of squares and efficient domination in graph classes" by T. Karthick
 

Structure of squares and efficient domination in graph classes

Article Type

Research Article

Publication Title

Theoretical Computer Science

Abstract

For any two vertices u and v in a graph G, dG(u,v) denote the distance between u and v in G. The square of a graph G=(V,E) is the graph G2=(V,E2) such that uv∈E2 if and only if dG(u,v)∈{1,2}. An efficient dominating set (e.d. for short) in G is a subset D of vertices such that D is an independent set and each vertex outside D has exactly one neighbor in D. In general, (i) if P is a (hereditary) graph property, and if a graph G satisfies P, then G2 does not need to satisfy P, and (ii) if H is any graph, and if G is a H-free graph that has an e.d., then G2 need not be H-free. In this paper, we show that the squares of (P6, banner)-free graphs that have an e.d. are again (P6, banner)-free. This result together with some known results implies that the EFFICIENT DOMINATING SET problem (which asks for the existence of an e.d. in a given graph G) can be solved in time O(n3) for (P6, banner)-free graphs. Here, Pt denotes the chordless path on t vertices, and a banner is the graph obtained from a chordless cycle on four vertices by adding a vertex that has exactly one neighbor on the cycle.

First Page

38

Last Page

46

DOI

10.1016/j.tcs.2016.09.002

Publication Date

11-1-2016

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