In [16] Nowakowski and Rall listed a series of conjectures involving several different graph products. particular, they conjectured that $$i(G\times H) \ge i(G)i(H)$$ where i(G) is the independent domination number G $$G\times H$$ direct product graphs H. We show this conjecture false, and, in fact, construct pairs for which $$\min \{i(G), i(H)\} - i(G\times H)$$ arbitrarily large. also give ex...

Let G = (V, E) be a graph. Set D ⊆ V (G) is a total outerconnected dominating set of G if D is a total dominating set in G and G[V (G)−D] is connected. The total outer-connected domination number of G, denoted by γtc(G), is the smallest cardinality of a total outer-connected dominating set of G. We show that if T is a tree of order n, then γtc(T ) ≥ d 2n 3 e. Moreover, we constructively charact...

Let G be an undirected graph with vertex and edge sets V (G) E(G), respectively. A set S ⊆ is called a hop independent dominating of if both G. The minimum cardinality G, denoted by γhih(G), the domination number In this paper, we show that lies between independence We characterize these types in shadow graph, join, corona, lexicographic product two graphs. Moreover, either exact values or boun...

In a graph G, a vertex dominates itself and its neighbors. A subset S of V is called a dominating set in G if every vertex in V-S is adjacent to at least one vertex in S. The minimum cardinality taken over all, the minimal double dominating set which is called Fuzzy Double Domination Number and which is denoted as ) (G fdd  . A set V S  is called a Triple dominating set of a graph G if every ...

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