Given a graphG = (V , E)with no isolated vertex, a subset S of V is called a total dominating set of G if every vertex in V is adjacent to a vertex in S. A total dominating set S is called a differentiating-total dominating set if for every pair of distinct vertices u and v in V , N[u] ∩ S ≠ N[v] ∩ S. The minimum cardinality of a differentiating-total dominating set of G is the differentiating-...

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set is the total domination number t(G). We show that for a nontrivial tree T of order n and with ` leaves, t(T ) > (n+2 `)=2, and we characterize the trees attaining this lower bound. Keywords: total domination, trees. AMS subject classi...

A set S V is a dominating set of a graph G = (V;E) if each vertex in V is either in S or is adjacent to a vertex in S. A vertex is said to dominate itself and all its neighbors. The domination number (G) is the minimum cardinality of a dominating set of G. In terms of a chess board problem, let Xn be the graph for chess pieceX on the square of side n. Thus, (Xn) is the domination number for che...

Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number γ×k(G) of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V , |NG[v]∩S| ≥ k. Also the total k-domination number γ×k,t(G) of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V , |NG(v)∩S| ≥ k. The k-transversal numb...

Let γ {k} t (G) denote the total {k}-domination number of graph G, and let G H denote the Cartesian product of graphs G and H . In this paper, we show that for any graphs G and H without isolated vertices, γ {k} t (G)γ {k} t (H) ≤ k(k + 1)γ {k} t (G H). As a corollary of this result, we have γt (G)γt (H) ≤ 2γt (G H) for all graphs G and H without isolated vertices, which is given by Pak Tung Ho...

In this paper, a necessary and sufficient condition for the existence of an efficient 2-dominating set in a class of circulant graphs has been obtained and for those circulant graphs, an upper bound for the 2domination number is also obtained. For the circulant graphs Cir(n,A), where A = {1, 2, . . . , x, n − 1, n − 2, . . . , n − x} and x ≤ bn−1 2 c, the perfect 2-tuple total domination number...

In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turán [8], we present a sharp lower bound on Kr+1-free graphs for r ≥ 2. Applying the concept of total limited packing we bound the signed total domination number of G with δ(G) ≥ 3 from above by n−2b 2ρo(G)+δ−3...

In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453–1462], where the players Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a t...

A subset S of vertices of a graph G with no isolated vertex is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex in V (G) − S is also adjacent to a vertex in V (G) − S. The total restrained domination number of G is the minimum cardinality of a total restrained dominating set of G. In this paper we initiate the study of total restrained bondage in g...