A connected domination set of a graph is a set of vertices such that every vertex not in is adjacent to and the induced subgraph of is connected. The minimum connected domination set of a graph is the connected domination set with the minimum number of vertices. In this paper, we propose parallel algorithms for finding the minimum connected domination set of interval graphs and circular-arc gra...

In this paper we introduce the concept of strong total domination in fuzzy graphs. We determine the strong total domination number for several classes of fuzzy graphs. A lower bound and an upper bound for the strong total domination number in terms of strong domination number is obtained. Strong total domination in fuzzy trees is studied. A necessary and sufficient condition for the set of fuzz...

We find the maximum number of edges for a graph of given order and value of parameter for several domination parameters. In particular, we consider the total domination and independent domination numbers.

The paired bondage number (total restrained bondage number, independent bondage number, k-rainbow bondage number) of a graph G, is the minimum number of edges whose removal from G results in a graph with larger paired domination number (respectively, total restrained domination number, independent domination number, k-rainbow domination number). In this paper we show that the decision problems ...

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...

For an orientation G of a simple graph G, -Ni7 [x] denotes the vertex x together with all those vertices in ~ for which there are arcs directed toward x. A set S of vertices of ~ is an efficient dominating set (EDS) of G provided that [IqS[x]nS [ = 1 for every x in G. An efficiency of G is an ordered pair (~, S), where S is an EDS of the orientation ~ of G. The number of distinct efficiencies o...

The paired-domination was introduced by Haynes and Slater in 1995 as a model for assigning backups to guards for security proposes, and it was shown that the problem on general graphs is NP-complete. In 2003, Qiao et al gave a linear algorithm for obtaining the paired-domination numbers of trees based on the paired-domination numbers of paths. The advantage of taking paths into consideration is...

Effective solutions to problems encountered in networks are often based on whether the elementary set can be partitioned into classes according to some specific criteria. The chromatic number of a graph G(V,E) is the minimum number of colours needed to colour the vertices of G such that no two adjacent vertices have the same colour. An independent set D of G is called an efficient dominating se...