Fractional edge domination in graphs

Abstract

Let G = (V,E) be a graph. A function f : E ? [0, 1] is called an edge dominating function if ?x?N[e] f(x)?1 for all e ? E(G), where N[e] is the closed neighbourhood of the edge e. An edge dominating function f is called minimal (MEDF) if for all functions g : E ? [0,1] with g < f, g is not an edge dominating function. The fractional edge domination number ?'f and the upper fractional edge domination number ?'f are defined by ?'f (G) = min{|f| : f is an MEDF of G} and ?'f (G) = max{|f| : f is an MEDF of G}, where |f| = ?e?E f(e). Further we introduce the fractional parameters corresponding to edge irredundance and edge independence, leading to the fractional edge domination chain. We also consider topological properties of the set of all edge dominating functions of G.