Fractional Independence and Fractional Domination Chain in Graphs

Abstract

Let G = (V, E) be a graph. A function f: V → [0,1] is called an independent function if for every vertex v with f (v) > 0, we have ∑u∊N[v] f(u) = 1. An independent function f is called a maximal independent function (MIF) if for every v ∊ V with f (v) = 0, we have ∑u∊N[v] f(u) ≥ 1. The fractional independent domination number if and the fractional independence number β0f are defined by if = min{|f| = ∑u∊V f(u): f is an MIF of G} and β0f = max{|f| |: f is an MIF of G}. These parameters along with the fractional irredundance numbers irf, IRf and the fractional domination numbers γf, Γf lead to the fractional domination chain given by irf ≤ γf ≤ if ≤ β0f ≤ Γf ≤ IRf. In this paper we present several results on the parameters if and β0f.