On total domination in the Cartesian product of graphs

Abstract

Ho proved in [A note on the total domination number, Util. Math. 77 (2008) 97-100] that the total domination number of the Cartesian product of any two graphs without isolated vertices is at least one half of the product of their total domination numbers. We extend a result of Lu and Hou from [Total domination in the Cartesian product of a graph and K 2 or C n , Util. Math. 83 (2010) 313-322] by characterizing the pairs of graphs G and H for which γ t (G H) = 1 2 γ t (G)γ t (H) , whenever γ t (H) = 2. In addition, we present an infinite family of graphs G n with γ t (G n ) = 2n, which asymptotically approximate the equality in γ t (G n G n ) ≥ 1 2 γ t (G n ) 2 .

Total domination in generalized prisms and a new domination invariant

“…Then by the definition of f , v is either adjacent to at least one neighbor u with |f (u)| = 3 (namely to a vertex u such that f (u) = {1, 2, 3}) or v has two neighbors u, w such that |f (u)| = |f (w)| = 2, or v has one neighbor u with |f (u)| = 2 and one neighbor w with |f (w)| = 1. In either of the three cases, we obtain that c(v) ≥ 4 5 . Now we can derive…”

“…Corollary 10. If G is a graph, then γ 2rt (G) ≥ 4 3 γ(G), and this bound is tight. By Corollary 8 the remaining interesting cases for cycles and paths are for k = 2 and k = 3.…”

“…• each vertex v with the initial weight 3, i.e., with the property |f (v)| = 3, contributes 4 5 to the new weight of each vertex from…”

“…Lu and Hou [18] characterized graphs H which satisfy γ t (H 2 K 2 ) = γ t (H) and γ t (C n )γ t (H) = 2γ t (C n 2 H), respectively. Brešar et al [4] extended their result by characterizing the pairs of graphs G and H for which 2γ t (G 2 H) = γ t (G)γ t (H), whenever γ t (H) = 2. A recent result of Azarija et al [3] that γ t (Q n+1 ) = 2γ(Q n ) holds for all n ≥ 1 follows from a more general result on bipartite prisms.…”

A Vizing-type result for semi-total domination

“…The breakthrough "double-projection" result of Clark and Suen [5] gave the first Vizingtype bound of γ(G ✷ H) ≥ 1 2 γ(G)γ(H). Recently, Brešar [1] improved this bound to γ(G ✷ H) ≥ (2γ(G)−ρ(G))γ(H) 3 , where ρ(G) is the two-packing number of G. For more on attempts to solve Vizing's conjecture over more than five decades since it was stated, see the survey [2].…”