3 research outputs found
On the integrity of domination in graphs.
Thesis (M.Sc.)-University of Natal, 1993.This thesis deals with an investigation of the integrity of domination in a.graph, i.e., the extent to
which domination properties of a graph are preserved if the graph is altered by the deletion of
vertices or edges or by the insertion of new edges.
A brief historical introduction and motivation are provided in Chapter 1. Chapter 2 deals with kedge-(
domination-)critical graphs, i.e., graphsG such that )'(G) = k and )'(G+e) < k for all e E
E(G). We explore fundamental properties of such graphs and their characterization for small
values of k. Particular attention is devoted to 3-edge-critical graphs.
In Chapter 3, the changes in domination number brought aboutby vertex removal are investigated.
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Parameters )'+'(G) (and "((G)), denoting the smallest number of vertices of G in a set 5 such that
)'(G-5) > )'(G) ()'(G -5) < )'(G), respectively), are investigated, as are'k-vertex-critical graphs G
(with )'(G) = k and )'(G-v) < k for all v E V(O)). The existence of smallest'domination-forcing
sets of vertices of graphs is considered.
The bondage number 'Y+'(G), i.e., the smallest number of edges of a graph G in a set F such that
)'(G- F) > )'(0), is investigated in Chapter 4, as are associated extremal graphs. Graphs with
dominating sets or domination numbers that are insensitive to the removal of an arbitrary edge are
considered, with particular reference to such graphs of minimum size.
Finally, in Chapter 5, we-discuss n-dominating setsD of a graph G (such that each vertex in G-D
is adjacent to at least n vertices in D) and associated parameters. All chapters but the first and
fourth contain a listing of unsolved problems and conjectures
Aspects of distance and domination in graphs.
Thesis (Ph.D.-Mathematics and Applied Mathematics)-University of Natal, 1995.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