Transversal Game on Hypergraphs and the $\frac{3}{4}$-Conjecture on the Total Domination Game

Abstract

The 3 4 -Game Total Domination Conjecture posed by Henning, Klavžar and Rall [Combinatorica, to appear] states that if G is a graph on n vertices in which every component contains at least three vertices, then γ tg (G) ≤ 3 4 n, where γ tg (G) denotes the game total domination number of G. Motivated by this conjecture, we raise the problem to a higher level by introducing a transversal game in hypergraphs. We define the game transversal number, τ g (H), of a hypergraph H, and prove that if every edge of H has size at least 2, and H C 4 , then τ g (H) ≤ 4 11 (n H + m H ), where n H and m H denote the number of vertices and edges, respectively, in H. Further, we characterize the hypergraphs achieving equality in this bound. As an application of this result, we prove that if G is a graph on n vertices with minimum degree at least 2, then γ tg (G) < 8 11 n. As a consequence of this result, the 3 4 -Game Total Domination Conjecture is true over the class of graphs with minimum degree at least 2.

Progress towards the 1/2-Conjecture for the domination game

“…To see this, we consider a variant of the domination game, which can be seen as a generalisation of most other variants that have been studied. The transversal game was defined in [3] as a game played on a hypergraph H, in which the two players, Edge-hitter and Staller, alternately select a vertex from H, with the rule that each newly selected vertex must hit, i.e., be contained in, at least one edge that does not intersect the set of previously selected vertices. The game ends when the set of selected vertices becomes a transversal in H, i.e., when every edge of H intersects the set of selected vertices.…”

“…In [10], the authors proved the following result. The closed neighbourhood hypergraph of a graph G is defined as hypergraph H G with vertex set V (H G ) = V (G) and hyperedge set E(H G ) = {N G [x]|x ∈ V (G)} consisting of the closed neighbourhoods of vertices in G. As remarked in [3], the domination game played on a graph G can be seen as a special instance of the transversal game being played on the closed neighbourhood hypergraph H G of G. By taking C = {H G : G is a graph of minimum degree at least 2} and c = 1/2, we obtain the following as a direct corollary of Lemma 3.4. Corollary 3.5.…”

The (a, b)-monochromatic transversal game on clique-hypergraphs of powers of cycles

“…Combinatorial games [4] are alternating finite two-player games of pure strategy in which all the relevant information is public to both players, as well as no randomness or luck is allowed. A combinatorial game using the concept of tranversals in hypergraphs is the transversal game, presented by Bujts et al [5,6]. In this game, two players, Edge-hitter and Staller, take turns choosing a vertex from H. Each chosen vertex must hit at least one edge not hit by the vertices previously chosen.…”

Transversals in hypergraphs through a new combinatorial game

“…By the point of view of combinatorial games [3], i.e., alternating finite two-player games of pure strategy in which all the relevant information is public to both players, as well as no randomness or luck are allowed, Butjás et al [4,5] presented a combinatorial game using the concept of tranversals in hypergraphs that was called transversal game. A transversal in a hypergraph is a subset of the vertex set that intersects every hyperedge [2].…”