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 = { w1 , w2 , [Special characters omitted.] &cdots; , wk } of vertices in G and a vertex v of G , the code of v with respect to W is the k -vector cW (v ) = (d ( v, w1 ),d (v, w 2 ), [Special characters omitted.] &cdots; , d (v, wk )). 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 Gis 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 , S2 , [Special characters omitted.] &cdots; , Sk } of V (G ) and a vertex v of G , the code of v with respect to Π is defined as the k -vector cΠ (v ) = (d ( v, S1 ), d (v, S 2 ), [Special characters omitted.] &cdots; , d (v, Sk )). 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 = e0 ,e1 , ..., ek = f of edges of G such that ei 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 = {G1 , G2 , ..., Gk } 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, G2 ), ..., d (e, Gk )). 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.

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