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...

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...

For a simple graph G=(V,E) with no isolated vertices, total Roman {3}-dominating function(TR3DF) on G is function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 f(v)=1; and (iii) every vertex v f(v)≠0 has neighbor u f(u)≠0 for v∈V(G). The weight of TR3DF f sum f(V)=∑v∈V(G)f(v) minimum called {3}-domination number denoted by γt{R3}(G). In this paper, we...

A graph is k-domination-critical if γ(G)=k, and for any edge e not in G, γ(G+e) = k-1. In this paper we show that the diameter of a domination k-critical graph with k≥2 is at most 2k-2. We also show that for every k≥2, there is a k-domination-critical graph having diameter 3 2 k 1 −     . We also show that the diameter of a 4-domination-critical graph is at most 5.

We settle two conjectures on domination-search, a game proposed by Fomin et.al. [1], one in affirmative and the other in negative. The two results presented here are (1) domination search number can be greater than domination-target number, (2) domination search number for asteroidal-triple-free graphs is at most 2.

We show that the diameter of a total domination vertex-critical graph is at most 5(γt −1)/3, and that the diameter of an independent domination vertex-critical graph is at most 2(i− 1). For all values of γt ≡ 2 (mod 3) there exists a total domination vertex-critical graph with the maximum possible diameter. For all values of i ≥ 2 there exists an independent domination vertex-critical graph wit...