Independent domination of graphs with bounded maximum degree

Abstract

An independent dominating set of a graph, also known as a maximal independent set, is a set S of pairwise non-adjacent vertices such that every vertex not in S is adjacent to some vertex in S. We prove that for Delta = 4 or Delta >= 6, every connected n -vertex graph of maximum degree at most Delta has an independent dominating set of size at most (1 - [Delta 2/4]+Delta )(n - 1) + 1. In addition, we characterize all connected graphs having the equality and we show that other connected graphs have an independent dominating set of size at most (1 -Delta/Delta(2)/4]+Delta )n. (c) 2022 Elsevier Inc. All rights reserved.