Independent domination of graphs with bounded maximum degree

TL;DR

This paper establishes upper bounds on the size of independent dominating sets in connected graphs with maximum degree  or , providing exact characterizations for extremal cases.

Contribution

It proves new bounds on independent dominating set sizes for graphs with bounded maximum degree and characterizes extremal graphs achieving equality.

Findings

Bound of  or  on independent dominating set size established

Characterization of all extremal graphs with equality

General bound for all connected graphs with maximum degree  or 

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 $\Delta\ge 6$, every connected $n$-vertex graph of maximum degree at most $\Delta$ has an independent dominating set of size at most $(1-\frac{\Delta}{\lfloor\Delta^2/4\rfloor+\Delta})(n-1)+1$. 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 $(1-\frac{\Delta}{ \lfloor\Delta^2/4\rfloor+\Delta})n$.