On graphs having one size of maximal open packings

TL;DR

This paper investigates graphs where all maximal open packings have the same size, providing a construction method and a structural characterization for certain graphs without small cycles.

Contribution

It introduces the class of graphs with uniform maximal open packing sizes, offers a construction method, and characterizes such graphs without small cycles.

Findings

Every graph is an induced subgraph of a graph with uniform maximal open packings.

Structural characterization of graphs without small cycles where the minimal and maximal open packing sizes coincide.

Main result applies to graphs with no cycles shorter than 15, where these invariants are equal.

Abstract

A set $P$ of vertices in a graph $G$ is an open packing if no two distinct vertices in $P$ have a common neighbor. Among all maximal open packings in $G$, the smallest cardinality is denoted $\rho^{\rm o}_L(G)$ and the largest cardinality is $\rho^{\rm o}(G)$. There exist graphs for which these two invariants are arbitrarily far apart. In this paper we begin the investigation of the class of graphs that have one size of maximal open packings. By presenting a method of constructing such graphs we show that every graph is the induced subgraph of a graph in this class. The main result of the paper is a structural characterization of those $G$ that do not have a cycle of order less than $15$ and for which $\rho^{\rm o}_L(G)=\rho^{\rm o}(G)$.