A cactus graph is a connected graph in which any two cycles have at most one vertex in common. Let γ(G) and γc(G) be the domination number and connected domination number of a graph G, respectively. We can see that γ(G) ≤ γc(G) for any graph G. S. Arumugam and J. Paulraj Joseph [1] have characterized trees, unicyclic graphs and cubic graphs with equal domination and connected domination numbers...

This paper consists of two loosely related notes on the domination number of graphs. In the first part, we provide a new upper bound for the domination number of d-regular graphs. Our bound is the best known for d ≥ 6. In the second part, we compute the exact domination number and total domination number of certain Kneser graphs, and we provide some bounds on the domination number of other Knes...

Upper and lower bounds on the total domination number of the direct product of graphs are given. The bounds involve the {2}-total domination number and the total 2-tuple domination number of the factors. Using these relationships some exact total domination numbers are obtained. An infinite family of graphs is constructed showing that the bounds are best possible. The domination number of direc...

In many papers, the relation between the domination number of a product of graphs and the product of domination numbers of factors is studied. Here we investigate this problem for domination and total domination numbers in the cross product of digraphs. We give analogues of known results for graphs, and we also present new results for digraphs with sources. Using these results we find dominatio...

In this paper, we consider various types of domination vertex critical graphs, including total domination vertex critical graphs and independent domination vertex critical graphs and connected domination vertex critical graphs. We provide upper bounds on the diameter of them, two of which are sharp. MSC (2010): 05C12, 05C69

A function f : V → {−1, 0, 1} is a minus-domination function of a graph G = (V,E) if the values over the vertices in each closed neighborhood sum to a positive number. The weight of f is the sum of f(x) over all vertices x ∈ V. The minus-domination number γ(G) is the minimum weight over all minus-domination functions. The size of a minus domination is the number of vertices that are assigned 1....

We briefly review known results about the signed edge domination number of graphs. In the case of bipartite graphs, the signed edge domination number can be viewed in terms of its bi-adjacency matrix. This motivates the introduction of the signed domination number of a (0, 1)-matrix. We investigate the signed domination number for various classes of (0, 1)-matrices, in particular for regular an...

In this paper we consider the effect of edge contraction on the domination number and total domination number of a graph. We define the (total) domination contraction number of a graph as the minimum number of edges that must be contracted in order to decrease the (total) domination number. We show both of this two numbers are at most three for any graph. In view of this result, we classify gra...

Upper and lower bounds on the total domination number of the direct product of graphs are given. The bounds involve the {2}-total domination number, the total 2-tuple domination number, and the open packing number of the factors. Using these relationships one exact total domination number is obtained. An infinite family of graphs is constructed showing that the bounds are best possible. The dom...

The domination problem and its variants have been extensively studied in the literature. In this paper we investigate the domination problem in distance-hereditary graphs. In particular, we give a linear-time algorithm for the domination problem in distance-hereditary graphs by a labeling approach. We actually solve a more general problem, called the L-domination problem, which also includes th...