6 research outputs found

    On the Independent Domination Number of Regular Graphs

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    A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. In this paper, we consider questions about independent domination in regular graphs

    Equitable total domination in graphs

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    A subset ‫ܦ‬ of a vertex set ܸሺ‫ܩ‬ሻ of a graph ‫ܩ‬ ൌ ሺܸ, ‫ܧ‬ሻ is called an equitable dominating set if for every vertex ‫ݒ‬ ‫א‬ ܸ െ ‫ܦ‬ there exists a vertex ‫ݑ‬ ‫א‬ ‫ܦ‬ such that ‫ݒݑ‬ ‫א‬ ‫ܧ‬ሺ‫ܩ‬ሻ and |݀݁݃ሺ‫ݑ‬ሻ െ ݀݁݃ሺ‫ݒ‬ሻ| 1, where ݀݁݃ሺ‫ݑ‬ሻ and ݀݁݃ሺ‫ݒ‬ሻ are denoted as the degree of a vertex ‫ݑ‬ and ‫ݒ‬ respectively. The equitable domination number of a graph ߛ ሺ‫ܩ‬ሻ of ‫ܩ‬ is the minimum cardinality of an equitable dominating set of ‫.ܩ‬ An equitable dominating set ‫ܦ‬ is said to be an equitable total dominating set if the induced subgraph ‫ۄܦۃ‬ has no isolated vertices. The equitable total domination number ߛ ௧ ሺ‫ܩ‬ሻ of ‫ܩ‬ is the minimum cardinality of an equitable total dominating set of ‫.ܩ‬ In this paper, we initiate a study on new domination parameter equitable total domination number of a graph, characterization is given for equitable total dominating set is minimal and also discussed Northaus-Gaddum type results

    Aspects of distance and domination in graphs.

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    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

    On the integrity of domination in graphs.

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    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. \ 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
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