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...

The six basic parameters relating to domination, independence and irredundance satisfy a chain of inequalities given by ir ≤ γ ≤ i ≤ β0 ≤ Γ ≤ IR where ir, IR are the irredundance and upper irredundance numbers, γ,Γ are the domination and upper domination numbers and i, β0 are the independent domination number and independence number respectively. In this paper, we introduce the concept of indep...

a dominating set $d subseteq v$ of a graph $g = (v,e)$ is said to be a connected cototal dominating set if $langle d rangle$ is connected and $langle v-d rangle neq phi$, contains no isolated vertices. a connected cototal dominating set is said to be minimal if no proper subset of $d$ is connected cototal dominating set. the connected cototal domination number $gamma_{ccl}(g)$ of $g$ is the min...

MacGillivray and Seyffarth (J. Graph Theory 22 (1996), 213–229) proved that planar graphs of diameter two have domination number at most three and planar graphs of diameter three have domination number at most ten. They also give examples of planar graphs of diameter four having arbitrarily large domination numbers. In this paper we improve on their results. We prove that there is in fact a uni...

In this paper, the new kind of parameter strong (weak) domination number in a bipolar fuzzy graph is defined and established the parametric conditions. Another new kind of parameter a totalstrong (weak) bipolar domination number is defined and established the parametric conditions. The properties of strong (weak) bipolar domination number and totalstrong (weak) bipolar domination numbersare dis...