Grundy dominating sequences and zero forcing sets

In a graph $G$ a sequence $v_1,v_2,\dots,v_m$ of vertices is Grundy dominating if for all $2\le i \le m$ we have $N[v_i]\not\subseteq \cup_{j=1}^{i-1}N[v_j]$ and is Grundy total dominating if for all $2\le i \le m$ we have $N(v_i)\not\subseteq \cup_{j=1}^{i-1}N(v_j)$. The length of the longest Grundy (total) dominating sequence has been studied by several authors. In this paper we introduce two similar concepts when the requirement on the neighborhoods is changed to $N(v_i)\not\subseteq \cup_{j=1}^{i-1}N[v_j]$ or $N[v_i]\not\subseteq \cup_{j=1}^{i-1}N(v_j)$. In the former case we establish a strong connection to the zero forcing number of a graph, while we determine the complexity of the decision problem in the latter case. We also study the relationships among the four concepts, and discuss their computational complexities

On graphs with small game domination number

International audienceThe domination game is played on a graph G by Dominator and Staller. The game domination number γg(G) of G is the number of moves played when Dominator starts and both players play optimally. Similarly, γ′g(G) is the number of moves played when Staller starts. Graphs G with γg(G)=2, graphs with γ′g(G)=2, as well as graphs extremal with respect to the diameter among these graphs are characterized. In particular, γ′g(G)=2 and diam(G)=3 hold for a graph G if and only if G is a so-called gamburger. Graphs G with γg(G)=3 and diam(G)=6, as well as graphs G with γ′g(G)=3 and diam(G)=5 are also characterize