New results on the 1-isolation number of graphs without short cycles

TL;DR

This paper establishes upper bounds on the 1-isolation number for certain graphs without short cycles, showing that the minimum 1-isolating set size is at most a quarter of the graph's order, with sharp bounds.

Contribution

It proves new bounds on the 1-isolation number for connected graphs without specific short cycles, extending understanding of graph isolation properties.

Findings

Bound of n/4 for graphs without 6-cycles or induced 5- and 6-cycles

Bounds are sharp for the specified classes of graphs

Results apply to connected graphs excluding certain small cycles

Abstract

Let $G$ be a graph. A subset $D \subseteq V(G)$ is called a 1-isolating set of $G$ if $\Delta(G-N[D]) \leq 1$, that is, $G-N[D]$ consists of isolated edges and isolated vertices only. The $1$-isolation number of $G$, denoted by $\iota_1(G)$, is the cardinality of a smallest $1$-isolating set of $G$. In this paper, we prove that if $G \notin \{P_3,C_3,C_7,C_{11}\}$ is a connected graph of order $n$ without $6$-cycles, or without induced 5- and 6-cycles, then $\iota_1(G) \leq \frac{n}{4}$. Both bounds are sharp.