Fractional global domination in graphs

Abstract

Let G = (V, E) be a graph. A function g : V → [0, 1] is called a global dominating function (GDF) of G, if for every v ∈ V, g(N[v]) = ∑u∈N[v] g(u) ≥ 1 and g(N(v)) = ∑u∉N(v) g(u) ≥ 1. A GDF g of a graph G is called minimal (MGDF) if for all functions f : V → [0, 1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number γfg(G) is defined as follows: γfg(G) = min{|g| : g is an MGDF of G} where |g| = ∑v∈V g(v). In this paper we initiate a study of this parameter.