Grundy dominating sequences and zero forcing sets

TL;DR

This paper introduces new variants of Grundy dominating sequences with altered neighborhood conditions, explores their connections to zero forcing sets, and analyzes their computational complexities.

Contribution

It defines new concepts of Grundy dominating sequences with modified neighborhood conditions, links one variant to zero forcing number, and determines the complexity of related decision problems.

Findings

Established a strong connection between one variant and the zero forcing number.

Determined the computational complexity of the decision problem for another variant.

Explored relationships and complexities among four new concepts of Grundy dominating sequences.

Abstract

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.