Dissertation Defense 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 Vf+  = {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) - Vf +, 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 ƴc(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) = ƴc(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, ƴc (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 ƴc(G) = ƴ(G) < rad G and infinitely many graphs G with ƴc(G) = rad G < ƴc (G). Those graphs G having ƴc(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 Vf+ 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. Related Topics Main List of Archives: Dissertation Defenses Current Dissertation Defenses
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