On the equivalence of the upper open irredundance and fractional upper open irredundance numbers of a graph

A set S ⊂ K of vertices in a graph G = (V, E) is called open irredundant if for every vertex ν ∈ S there exists a vertex w ∈ V S such that w is adjacent to ν 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 → [0,1] is called open irredundant if for every vertex ν ∈ V, g(ν) > 0 implies there exists a vertex w adjacent to ν 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 ≠ h and g(ν) ≤ h(ν), for every ν S V. The fractional upper open irredundance number equals OIRf(G) = sup{|g| : g is an open irredundant function on G}. In this paper we prove that for any graph G, OIR(G) = OIRf(G).