No description available.

No description available.

No description available.

No description available.

Graph-monoids are introduced as algebraic objects which correspond to congruences over graphs. Varieties are defined, and a one-to-one correspondence between varieties of graph-monoids and varieties of graph-congruences is demonstrated. This is viewed as a generalization of the known correspondence between monoid varieties and congruence varieties. Some classification and effective characterizations of graph-monoid varieties are obtained. These are used to decide membership in the wreath product of certain monoids varieties and the variety of definite semigroups.

No description available.

No description available.

This dissertation details the results of an investigation into, primarily, three aspects of graph vulnerability namely, l-connectivity, Steiner Distance hereditatiness and functional isolation. Following the introduction in Chapter one, Chapter two focusses on the l-connectivity of graphs and introduces the concept of the strong l-connectivity of digraphs. Bounds on this latter parameter are investigated and then the l-connectivity function of particular types of graphs, namely caterpillars and complete multipartite graphs as well as the strong l-connectivity function of digraphs, is explored. The chapter concludes with an examination of extremal graphs with a given l-connectivity. Chapter three investigates Steiner distance hereditary graphs. It is shown that if G is 2-Steiner distance hereditary, then G is k-Steiner distance hereditary for all k≥2. Further, it is shown that if G is k-Steiner distance hereditary (k≥ 3), then G need not be (k - l)-Steiner distance hereditary. An efficient algorithm for determining the Steiner distance of a set of k vertices in a k-Steiner distance hereditary graph is discussed and a characterization of 2-Steiner distance hereditary graphs is given which leads to an efficient algorithm for testing whether a graph is 2-Steiner distance hereditary. Some general properties about the cycle structure of k-Steiner distance hereditary graphs are established and are then used to characterize 3-Steiner distance hereditary graphs. Chapter four contains an investigation of functional isolation sequences of supply graphs. The concept of the Ranked supply graph is introduced and both necessary and sufficient conditions for a sequence of positive nondecreasing integers to be a functional isolation sequence of a ranked supply graph are determined. / Thesis (Ph.D.)-University of Natal, 1994.

The first half of this thesis deals with an aspect of domination; more specifically, we investigate the vertex integrity of n-distance-domination in a graph, i.e., the extent to which n-distance-domination properties of a graph are preserved by the deletion of vertices, as well as the following: Let G be a connected graph of order p and let oi- S s;:; V(G). An S-n-distance-dominating set in G is a set D s;:; V(G) such that each vertex in S is n-distance-dominated by a vertex in D. The size of a smallest S-n-dominating set in G is denoted by I'n(S, G). If S satisfies I'n(S, G) = I'n(G), then S is called an n-distance-domination-forcing set of G, and the cardinality of a smallest n-distance-domination-forcing set of G is denoted by On(G). We investigate the value of On(G) for various graphs G, and we characterize graphs G for which On(G) achieves its lowest value, namely, I'n(G), and, for n = 1, its highest value, namely, p(G). A corresponding parameter, 1](G), defined by replacing the concept of n-distance-domination of vertices (above) by the concept of the covering of edges is also investigated. For k E {a, 1, ... ,rad(G)}, the set S is said to be a k-radius-forcing set if, for each v E V(G), there exists Vi E S with dG(v, Vi) ~ k. The cardinality of a smallest k-radius-forcing set of G is called the k-radius-forcing number of G and is denoted by Pk(G). We investigate the value of Prad(G) for various classes of graphs G, and we characterize graphs G for which Prad(G) and Pk(G) achieve specified values. We show that the problem of determining Pk(G) is NP-complete, study the sequences (Po(G),Pl(G),P2(G), ... ,Prad(G)(G)), and we investigate the relationship between Prad(G)(G) and Prad(G)(G + e), and between Prad(G)(G + e) and the connectivity of G, for an edge e of the complement of G. Finally, we characterize integral triples representing realizable values of the triples b,i,p), b,l't,i), b,l'c,p), b,l't,p) and b,l't,l'c) for a graph. / Thesis (Ph.D.-Mathematics and Applied Mathematics)-University of Natal, 1995.

This thesis is a study of the symmetry of graphs and digraphs by considering certain homogeneous embedding requirements. Chapter 1 is an introduction to the chapters that follow. In Chapter 2 we present a brief survey of the main results and some new results in framing number theory. In Chapter 3, the notions of frames and framing numbers is adapted to digraphs. A digraph D is homogeneously embedded in a digraph H if for each vertex x of D and each vertex y of H, there exists an embedding of D in H as an induced subdigraph with x at y. A digraph F of minimum order in which D can be homogeneously embedded is called a frame of D and the order of F is called the framing number of D. We show that that every digraph has at least one frame and, consequently, that the framing number of a digraph is a well defined concept. Several results involving the framing number of graphs and digraphs then follow. Analogous problems to those considered for graphs are considered for digraphs. In Chapter 4, the notions of edge frames and edge framing numbers are studied. A nonempty graph G is said to be edge homogeneously embedded in a graph H if for each edge e of G and each edge f of H, there is an edge isomorphism between G and a vertex induced subgraph of H which sends e to f. A graph F of minimum size in which G can be edge homogeneously embedded is called an edge frame of G and the size of F is called the edge framing number efr(G) of G. We also say that G is edge framed by F. Several results involving edge frames and edge framing numbers of graphs are presented. For graphs G1 and G2 , the framing number fr(G1 , G2 ) (edge framing number ef r(GI, G2 )) of G1 and G2 is defined as the minimum order (size, respectively) of a graph F such that Gj (i = 1,2) can be homogeneously embedded in F. In Chapter 5 we study edge framing numbers and framing number for pairs of cycles. We also investigate the framing number of pairs of directed cycles. / Thesis (Ph.D.)-University of Natal, Pietermaritzburg, 1996.