In this paper we present eÆcient algorithms to solve domination set problem for the class of bounded tolerance graphs. The class of bounded tolerance graphs includes interval graphs and permtation graph as subclasses. Our solution to domination set problem has improved bounds for time complexity than the algorithm for the general co-comparability graphs which can as well be used for bounded tol...

Vizing’s conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained an...

We study the influence of edge subdivision on the convex domination number. We show that in general an edge subdivision can arbitrarily increase and arbitrarily decrease the convex domination number. We also find some bounds for unicyclic graphs and we investigate graphs G for which the convex domination number changes after subdivision of any edge in G.

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...

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. We call these graphs γt-critical. In this paper, we disprove a conjecture posed in a recent paper(On an open problem concerning total domination critical graphs, Expo. Mat...

We consider the domination number for on-line social networks, both in a stochastic network model, and for real-world, networked data. Asymptotic sublinear bounds are rigorously derived for the domination number of graphs generated by the memoryless geometric protean random graph model. We establish sublinear bounds for the domination number of graphs in the Facebook 100 data set, and these bou...

Bipartite graphs with equal edge domination number and maximum matching cardinality are characterized. These two parameters are used to develop bounds on the vertex cover and total vertex cover numbers of graphs and a resulting chain of vertex covering, edge domination, and matching parameters is explored. In addition, the total vertex cover number is compared to the total domination number of ...

Abstract. In this paper, connected domination in fuzzy graphs using strong arcs is introduced. The strong connected domination number of different classes of fuzzy graphs is obtained. An upper bound for the strong connected domination number of fuzzy graphs is obtained. Strong connected domination in fuzzy trees is studied. It is established that the set of fuzzy cut nodes of a fuzzy tree is a ...