A characterization of unicyclic graphs with the same independent domination number

Abstract

A set D of vertices of G is an independent dominating set if no two vertices of D are adjacent and every vertex not in D is adjacent to at lest one vertex in D . The independent domination number of a graph G , denoted by i ( G ), is the minimum cardinality of an independent dominating set in G . A unicyclic graph is a connected graph containing exactly one cycle. For k ≥ 1, let H ( k ) be the set of unicyclic graphs H satisfying i ( H ) = k . In this paper, we provide a constructive characterization of H ( k ) for all k ≥ 1.