Maker-Breaker total domination game

Maker-Breaker total domination game in graphs is introduced as a natural counterpart to the Maker-Breaker domination game recently studied by Duch\^ene, Gledel, Parreau, and Renault. Both games are instances of the combinatorial Maker-Breaker games. The Maker-Breaker total domination game is played on a graph $G$ by two players who alternately take turns choosing vertices of $G$. The first player, Dominator, selects a vertex in order to totally dominate $G$ while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. It is shown that there are infinitely many connected cubic graphs in which Staller wins and that no minimum degree condition is sufficient to guarantee that Dominator wins when Staller starts the game. An amalgamation lemma is established and used to determine the outcome of the game played on grids. Cacti are also classified with respect to the outcome of the game. A connection between the game and hypergraphs is established. It is proved that the game is PSPACE-complete on split and bipartite graphs. Several problems and questions are also posed.Comment: 21 pages, 5 figure

Disjoint Paired-Dominating sets in Cubic Graphs

A paired-dominating set of a graph G is a dominating set D with the additional requirement that the induced subgraph G[D] contains a perfect matching. We prove that the vertex set of every claw-free cubic graph can be partitioned into two paired-dominating sets

Partitioning the Vertices of a Cubic Graph Into Two Total Dominating Sets

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study cubic graphs whose vertex set can be partitioned into two total dominating sets. There are infinitely many examples of connected cubic graphs that do not have such a vertex partition. In this paper, we show that the class of claw-free cubic graphs has such a partition. For an integer k at least 3, a graph is k-chordal if it does not have an induced cycle of length more than k. Chordal graphs coincide with 3-chordal graphs. We observe that for k≥6, not every graph in the class of k-chordal, connected, cubic graphs has two vertex disjoint total dominating sets. We prove that the vertex set of every 5-chordal, connected, cubic graph can be partitioned into two total dominating sets. As a consequence of this result, we observe that this property also holds for a connected, cubic graph that is chordal or 4-chordal. We also prove that cubic graphs containing a diamond as a subgraph can be partitioned into two total dominating sets