On graphs having one size of maximal open packings

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 $ρ^{\rm o}_L(G)$ and the largest cardinality is $ρ^{\rm o}(G)$. 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 $ρ^{\rm o}_L(G)=ρ^{\rm o}(G)$.

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