In this paper we continue the investigation of total domination in Cartesian products of graphs first studied in Graphs Combin. 21 (2005), 63–69. A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γt(G). We prove ...

Let G be a graph with vertex set V ( ). A total Italian dominating function (TIDF) on is f : ) → {0, 1, 2} such that (i) every v = 0 adjacent to u 2 or two vertices w and z (ii) ≥ 1 1. The domination number γ tI the minimum weight of function. In this paper, we present Nordhaus–Gaddum type inequalities for number.

For a graph G = (V,E), a set D ⊆ V (G) is a total restrained dominating set if it is a dominating set and both 〈D〉 and 〈V (G)−D〉 do not have isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. A set D ⊆ V (G) is a restrained dominating set if it is a dominating set and 〈V (G) − D〉 does not contain an isolated vertex. Th...

Domination parameters in random graphs G(n, p), where p is a fixed real number in (0, 1), are investigated. We show that with probability tending to 1 as n → ∞, the total and independent domination numbers concentrate on the domination number of G(n, p).

In this paper, we continue the study of neighborhood total domination in graphs first studied by Arumugam and Sivagnanam [S. Arumugam, C. Sivagnanam, Neighborhood total domination in graphs, Opuscula Math. 31 (2011) 519–531]. A neighborhood total dominating set, abbreviated NTD-set, in a graph G is a dominating set S in G with the property that the subgraph induced by the open neighborhood of t...

For a graph G = (V,E), a subset D ⊆ V (G) is a total dominating set if every vertex of G has a neighbor in D. The total domination number of G is the minimum cardinality of a total dominating set of G. A subset D ⊆ V (G) is a 2-dominating set of G if every vertex of V (G) \ D has at least two neighbors in D, while it is a 2-outer-independent dominating set of G if additionally the set V (G) \ D...

The total domination number ? t ( G ) of a graph is the cardinality smallest set D ? V such that each vertex has neighbor in . annihilation largest integer k there exist different vertices with degree sum at most m It conjectured ? + 1 holds for every nontrivial connected conjecture been proved graphs minimum least 3, trees, certain tree-like graphs, block and cactus graphs. In main result this...