## Dissertations

### Cost Domination in Graphs

6-2001

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

Dr. Clifton Ealy

Dr. Michael Raines

Dr. Ping Zhang

#### 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 Vj~ = {v 6 V(G) : f(v) > 0} of f-dominating vertices is the f-dominating set. An /-dominating vertex v is said to f-dominate every vertex u with d(n, v) < f(u ), 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 cr(f) = lL,vev{G) f ( v )• The cost domination number 7 C(G) is the minimum value of cr(f) overall cost dominating functions f on G and a cost dominating function f with 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, j c(G) < m in{7 (G ),rad G}, where 7 (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 7 C(G) = 7 (G) < rad G and infinitely m any graphs G with 7 C(G) = rad G < 7 (G). Those graphs G having 7 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 E V (G) and (ii) g{u) < f ( u ) for some u E 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 cr(/) 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 Vj~ 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 cr(f) overall cost independent functions f , is investigated.

#### Access Setting

Dissertation-Open Access

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