Partial domination - the isolation number of a graph

TL;DR

This paper establishes a bound on the size of a vertex set in connected graphs that ensures the remaining vertices form an independent set, advancing the understanding of partial domination with structural constraints.

Contribution

It proves a sharp bound on the size of a dominating set that isolates the remaining vertices as an independent set in connected graphs, a novel result in partial domination.

Findings

Existence of a dominating set with size at most n/3 in connected graphs

Remaining vertices form an independent set after domination

Bound is proven to be sharp

Abstract

We prove the following result: If $G$ be a connected graph on $n \ge 6$ vertices, then there exists a set of vertices $D$ with $|D| \le \frac{n}{3}$ and such that $V(G) \setminus N[D]$ is an independent set, where $N[D]$ is the closed neighborhood of $D$. Furthermore, the bound is sharp. This seems to be the first result in the direction of partial domination with constrained structure on the graph induced by the non-dominated vertices, which we further elaborate in this paper.