Partition strategies for the Maker-Breaker domination game

TL;DR

It is shown that Dominator always wins in regular graphs and that deciding whether Dominator has a winning strategy as a second player can be computed in polynomial time for outerplanar and block graphs and implies that deciding whether Dominator has a winning strategy playing second is in NP for interval graphs.

Abstract

The Maker-Breaker domination game is a positional game played on a graph by two players called Dominator and Staller. The players alternately select a vertex of the graph that has not yet been chosen. Dominator wins if at some point the vertices she has chosen form a dominating set of the graph. Staller wins if Dominator cannot form a dominating set. Deciding if Dominator has a winning strategy has been shown to be a PSPACE-complete problem even when restricted to chordal or bipartite graphs. In this paper, we consider strategies for Dominator based on partitions of the graph into basic subgraphs where Dominator wins as the second player. Using partitions into cycles and edges (also called perfect [1,2]-factors), we show that Dominator always wins in regular graphs and that deciding whether Dominator has a winning strategy as a second player can be computed in polynomial time for outerplanar and block graphs. We then study partitions into subgraphs with two universal vertices, which is equivalent to considering the existence of pairing dominating sets with adjacent pairs. We show that in interval graphs, Dominator wins if and only if such a partition exists. In particular, this implies that deciding whether Dominator has a winning strategy playing second is in NP for interval graphs. We finally provide an algorithm in $n^{k+3}$ for $k$-nested interval graphs (i.e. interval graphs with at most $k$ intervals included one in each other).

Figures (6)

• Figure 1: A graph having a partition into three cliques of size at least $2$: the outcome is $\mathcal{D}$.
• Figure 2: The set $\{(u_1,v_1),(u_2,v_2),(u_3,v_3)\}$ is a pairing dominating set.
• Figure 3: Small graphs of outcome $\mathcal{D}$ used for covering strategies.
• Figure 4: Examples of graphs for which $\Delta = \delta + 1$ and Dominator loses as the second player.
• Figure 5: Both $F_1$ and $F_2$ are maximal $[1,2]$-partial factors of $G$. However, only $F_2$ is a cut-factor of $G$ as $i$, the only vertex not in $F_1$, is not adjacent to a cut-vertex in $G$ but $c$, the only vertex not in $F_2$, is adjacent to $d$ which is a cut-vertex. Moreover, in $G-d$ the vertex $e$, the neighbor of $d$ in $F_2$, is not in the same connected component as $c$.

Theorems & Definitions (66)

• Proposition 2.1: MBdomgame
• Proposition 2.2: Monotonicity MBdomgame
• Definition 2.3: MBdomgame
• Proposition 2.4: MBdomgame
• Lemma 2.5: Super lemma nacimthesis
• Lemma 2.6
• proof
• Definition 2.7
• Proposition 2.8
• proof