Grundy dominating sequences and zero forcing sets
Boštjan Brešar, Csilla Bujtás, Tanja Gologranc, Sandi Klavžar, Gašper Košmrlj, Balázs Patkós, Źsolt Tuza, Máté Vizer · Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) · 2017
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 $N[v_i] ot\subseteq \cup_{j=1}^{i-1}N[v_j]$ and is Grundy total dominating if for all $2\le i \le m$ we have $N(v_i) ot\subseteq \cup_{j=1}^{i-1}N(v_j)$. 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 $N(v_i) ot\subseteq \cup_{j=1}^{i-1}N[v_j]$ or $N[v_i] ot\subseteq \cup_{j=1}^{i-1}N(v_j)$. 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.