Candidate: David J. Erwin

Degree of: Doctor of Philosophy

Department: Mathematics and Statistics

Title: Cost Domination in Graphs

Date: Thursday, April 12, 2001, 4:00pm - 6:00pm, 6625 Everett Tower, Alavi Commons Room

Committee:
Dr. Gary Chartrand, Chair
Dr. Clifton Ealy
Dr. Michael Raines
Dr. Phing Zhang
Dr. Garry Johns

Abstract:
Let G be a connected graph having order at least 2.  A function f : V(G) → {0,1...,diam G} for which f (v) ≤ e (v) for every vertex v of G is a cost function on G.  A vertex v with f(v) > 0 is an f-dominating vertex, and the set V_f⁺  = {v ∈ V(G) : f(v) > 0} of f-dominating vertices is the f-dominating set. An f-dominating vertex v is said to f-dominate every vertex u with d(u,v) < f(v), while the vertices in V(G) - V_f ⁺, namely, those vertices of G that are not f-dominating, do not f-dominate any vertices of G. A cost dominating function on G is a cost function f in which every vertex is f-dominated by some vertex in the f-dominating set.

For a cost function f on a nontrivial connected graph G, let σ(f) = ∑_v∈V(G) f(v). The cost domination number ƴ_c(G) is the minimum value of σ(f) over all cost dominating functions f on G and a cost dominating function f with σ(f) = ƴ_c(G) is a minimum cost dominating function.

We establish several sharp upper and lower bounds on the cost domination number of a graph in terms of other well-known invariants. For example, ƴ_c (G) < min{<ƴ (G),rad G}, where ƴ(G) is the domination number of G and rad G is the radius of G. It is shown that there exist infinitely many graphs G with ƴ_c(G) = ƴ(G) < rad G and infinitely many graphs G with ƴ_c(G) = rad G < ƴ_c (G). Those graphs G having ƴ_c(G) < 3 are determined.

A cost dominating function f is minimal if there is no cost dominating function g satisfying (i) g(v) < f(v) for all v ∈ V(G) and (ii) g(u) < f(u) for some u ∈ V(G). The structure of the f-dominating set for both minimal and minimum cost dominating functions is determined. The upper cost domination number, which is the maximum value of σ(f) over all minimal cost dominating functions f on G, is also studied.

A cost function f is cost independent if there is no pair u, v of distinct vertices in V_f⁺ such that u is f-dominated by v. It is proved that for every graph G, there is a cost function on G that is both minimum cost dominating and cost independent. The cost independence number, which is the maximum value of σ(f) over all cost independent functions f, is investigated.

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