On the Equivalence of the Upper Open Irredundance and Fractional Upper Open Irredundance Numbers of a Graph

Abstract

A set $S\subset V$ of vertices in a graph $G=(V,E)$ is called open irredundant if for every vertex $v\in S$ there exists a vertex $w\in V\setminus S$ such that $w$ is adjacent to $v$ but to no other vertex in $S$. The upper open irredundance number $OIR(G)$ equals the maximum cardinality of an open irredundant set in $G$. A real-valued function $g:V\to [0,1]$ is called open irredundant if for every vertex $v\in V$, $g(v)>0$ implies there exists a vertex $w$ adjacent to $v$ such that $g(N[w])=1$. An open irredundant function $g$ is maximal if there does not exist an open irredundant function $h$ such that $g\ne h$ and $g(v)\le h(v)$, for every $v\in V$. The fractional upper open irredundance number equals $OI{R}_f(G)=sup\{|g|:g\text{ is an open irredundant function on }G\}$. In this paper we prove that for any graph $G$, $OIR(G)=OI{R}_f(G)$.