## Date of Award

12-2002

## Degree Name

Doctor of Philosophy

## Department

Mathematics

## Abstract

The distance *d* (*u, v* ) between two vertices *u* and *v* in a connected graph *G* is the length of a shortest *u* - *v* path in *G* . For an ordered set *W* = {* w*1 , *w*2 , [Special characters omitted.] &cdots; , *w**k* } of vertices in *G* and a vertex *v* of *G* , the code of *v* with respect to *W* is the *k* -vector *c**W* (*v* ) = (*d* (* v, w*1 ),*d* (*v, w* 2 ), [Special characters omitted.] &cdots; , *d* (*v, w**k* )). The set *W* is a resolving set for *G* if distinct vertices have distinct codes. A resolving set containing a minimum number of vertices is a basis for *G* . The dimension dim(*G* ) is the number of vertices in a basis for *G* . A resolving set *W* of *G*is connected if the subgraph [left angle bracket]* W* [right angle bracket] induced by *W* is a connected subgraph of *G* . The minimum cardinality of a connected resolving set *W* in a graph *G* is the connected resolving number *cr* (*G* ). A connected resolving set of cardinality *cr* (*G* ) is called a *cr* -set of *G* . We study the relationships between *cr* -sets and bases in a nontrivial connected graph *G* . The connected resolving numbers of some well-known classes of graphs are determined. It is shown, for a pair *a, b* of integers with 1 ≤ *a* ≤ *b* , that there exists a connected graph *G* with dim(* G* ) = *a* and *cr* (*G* ) = *b* if and only if (*a, b* ) ∉ {(1, *k*) : *k* ≥ 2}. For a triple *a, b, n* of integers with 1 ≤ *a* ≤ *b* ≤ *n* , there exists a connected graph *G* of order *n* such that dim(*G* ) = *a* and *cr* (*G* ) = *b* if and only if (i) *a* = *b* = 1, (ii) *a* ∈ {* n* - 2, *n* - 1} and *b* = *n* - 1, or (iii) 2 ≤ *a* ≤ *b* ≤ *n* - 2. We show that there exists a graph with a unique *cr* -set of cardinality *k* for every integer *k* ≥ 2. Minimal and forcing connected resolving sets in graphs are studied. A connected graph *H* is a resolving graph if there is a graph *G* with a minimum connected resolving set *W* such that the subgraph [left angle bracket]* W* [right angle bracket] of *G* induced by *W* is isomorphic to *H* . It is shown that every connected graph is a resolving graph.

For a set *S* of vertices in a connected graph *G* and a vertex *v* of *G* , the distance *d* (*v, S* ) between *v* and *S* is defined as *d* (*v, S* ) = min {* d* (*v, x* ): *x* ∈ *S* }. For an ordered *k* -partition Π = {*S* 1 , *S*2 , [Special characters omitted.] &cdots; , *S**k* } of *V* (*G* ) and a vertex *v* of *G* , the code of *v* with respect to Π is defined as the *k* -vector *c*Π (*v* ) = (*d* (* v, S*1 ), *d* (*v, S* 2 ), [Special characters omitted.] &cdots; , *d* (*v, S**k* )). The partition Π is called a resolving partition for *G* if distinct vertices of *G* have distinct codes with respect to Π. Resolving partitions with prescribed properties are studied.

For edges *e* and *f* in a connected graph *G* , the distance *d* (*e, f* ) between *e* and *f* is the minimum nonnegative integer *k* for which there exists a sequence *e* = *e*0 ,*e*1 , ..., *e**k* = *f* of edges of *G* such that *e**i* and *e** i*+1 are adjacent for *i* = 0, 1, ..., *k* - 1. For an edge *e* of *G* and a subgraph *F* of *G* , the distance between *e* and *F* is *d* (* e, F*) = min{*d* (*e, f* ) : *f* ∈ *E* (*F* )}. For an ordered *k* -decomposition [Special characters omitted.] D = {*G*1 , *G*2 , ..., *G**k* } of a connectedgraph *G* and *e* ∈ *E* (*G* ), the [Special characters omitted.] D -code of *e* is the *k* -vector [Special characters omitted.] cD (*e* ) = (*d* (*e, G* 1 ), *d* (*e, G*2 ), ..., *d* (*e, G**k* )). The decomposition [Special characters omitted.] D is a resolving decomposition for *G* if every two distinct edges of *G* have distinct [Special characters omitted.] D -codes. Resolving decompositions with prescribed properties are investigated.

## Access Setting

Dissertation-Open Access

## Recommended Citation

Saenpholphat, Varaporn, "Resolvability in Graphs" (2002). *Dissertations*. 1301.

https://scholarworks.wmich.edu/dissertations/1301