Effects on fractional domination in graphs

Abstract

A graph G is an undirected finite connected graph. A function f : V (G) → [0, 1] is called a fractional dominating function if, ∑u∈N[v]f (u) ≥1, for all v ∈ V, where N [v] is the closed neighborhood of v. The weight of a fractional dominating function is w (f) = ∑v∈V(G)f (v). The fractional domination number γf (G) has the least weight of all the fractional dominating functions of G. In this paper, we analyze the effects on γf (G) of deleting a vertex from G. Additionally, some bounds on γf (G) are discussed, and provide the exactness of some bounds. If we remove any leaves from any tree T, then the resulting graphs are , where |l| is the number of leaves. Some of the results are proved by the eccentricity value of a vertex e (v).