#### Date of Award

6-2001

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Clifton Ealy

#### Second Advisor

Dr. Michael Raines

#### Third Advisor

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

#### Recommended Citation

Erwin, David John, "Cost Domination in Graphs" (2001). *Dissertations*. 1365.

http://scholarworks.wmich.edu/dissertations/1365