A note on lower bounds for the total domination number of digraphs

Packing and domination parameters in digraphs

“…This implies the equality. Suppose now that the equality holds and that S is a γ o (D)-set in D. Then, the inequalities in (5) and (6) Hao and Chen [11] introduced the out-Slater number sℓ + (D) of a digraph D of order n as min{k | ⌊k/2⌋ + d + 1 + · · · + d + k ≥ n}, where d + 1 , · · · , d + k are the first k largest out-degrees of D. Among other results, they showed that γ t (D) ≥ sℓ + (D),…”

Integer domination of Cartesian product graphs

“…This "Vizing-like" inequality immediately suggested similar inequalities for total [8] and paired [9] domination (2008 and 2010, respectively). In 2011, Choudhary, Margulies and Hicks [3] improved the inequalities from [8,9] for total and paired domination by applying techniques similar to those of Clark and Suen, and also specific properties of binary matrices. In this paper, we explore integer domination (or {k}-domination), and again generate an improved inequality with this combined technique.…”

A Vizing-type result for semi-total domination

“…For example, Brešar, Henning, and Rall [4] defined the paired and rainbow domination numbers, and Henning and Rall [12] conjectured a Vizing-type inequality for total domination. This last conjecture was proved by Ho [14], who showed that for any graphs G and H, γ t (G ✷ H) ≥ 1 2 γ t (G)γ t (H). In this result, γ t (G) is the total domination number of G, which is the minimum size of a set T of vertices so that every vertex of G is adjacent to some vertex in T .…”