Justin Southey

Edge Weighting Functions on Dominating Sets

In this article, we use edge weighting functions on dominating sets to show that if we impose a regularity condition on a graph, then upper bounds on both the upper domination number and the upper total domination number can be greatly improved. More precisely, we prove that for k≥1 if G is a k-regular graph on n vertices, then the upper domination number of G is at most n/2, and the upper total domination number of G is at most n/(2−1k). Furthermore, we show that these bounds are sharp and characterize the infinite families of graphs that achieve equality in both these bounds.

Dominating and total dominating partitions in cubic graphs

In this paper, we continue the study of domination and total domination in cubic graphs. It is known [Henning M.A., Southey J., A note on graphs with disjoint dominating and total dominating sets, Ars Combin., 2008, 89, 159–162] that every cubic graph has a dominating set and a total dominating set which are disjoint. In this paper we show that every connected cubic graph on nvertices has a total dominating set whose complement contains a dominating set such that the cardinality of the total dominating set is at most (n+2)/2, and this bound is essentially best possible.

Nordhaus–Gaddum bounds for total domination

In the metadata of the chapter that will be visualized online, please replace the abstract with the following: “In this chapter, we consider Nordhaus- Gaddum inequalities involving the total domination number”

Disjoint dominating and total dominating sets in graphs

It has been shown [M.A. Henning, J. Southey, A note on graphs with disjoint dominating and total dominating sets, Ars Combin. 89 (2008) 159-162] that every connected graph with minimum degree at least two that is not a cycle on five vertices has a dominating set D and a total dominating set T which are disjoint. We characterize such graphs for which D@?T necessarily contains all vertices of the graph and that have no induced cycle on five vertices.

Independent Domination in Cubic Graphs

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 ini¾?S. The independent domination number of G, denoted byi¾?iG, is the minimum cardinality of an independent dominating set. In this article, we show that if Gi¾?C5i¾?K2 is a connected cubic graph of orderi¾?n that does not have a subgraph isomorphic to K2, 3, then iGi¾?3n/8. As a consequence of our main result, we deduce Reeds important result [Combin Probab Comput 5 1996, 277-295] that if G is a cubic graph of orderi¾?n, then γGi¾?3n/8, where γG denotes the domination number of G.

A characterization of the non-trivial diameter two graphs of minimum size

Let G be a graph with diameter two, order n and size m , such that no vertex is adjacent to every other vertex. A classical result due to Erd?s and Renyi (1962) states that m ? 2 n - 5 . We characterize the graphs that achieve equality in the Erd?s-Renyi bound.

Graphs with disjoint dominating and paired-dominating sets

A dominating set of a graph is a set of vertices such that every vertex not in the set is adjacent to a vertex in the set, while a paired-dominating set of a graph is a dominating set such that the subgraph induced by the dominating set contains a perfect matching. In this paper, we show that no minimum degree is sufficient to guarantee the existence of a disjoint dominating set and a paired-dominating set. However, we prove that the vertex set of every cubic graph can be partitioned into a dominating set and a paired-dominating set.