Dominating sequences under atomic changes with applications in Sierpi\'{n}ski and interval graphs

Abstract

A sequence $S=(v_1,\ldots,v_k)$ of distinct vertices of a graph $G$ is called a legal sequence if $N[v_i] \setminus \cup_{j=1}^{i-1}N[v_j]\not=\emptyset$ for any $i$. The maximum length of a legal (dominating) sequence in $G$ is called the Grundy domination number $\gamma_{gr}(G)$ of a graph $G$. It is known that the problem of determining the Grundy domination number is NP-complete in general, while efficient algorithm exist for trees and some other classes of graphs. In this paper we find an efficient algorithm for the Grundy domination number of an interval graph. We also show the exact value of the Grundy domination number of an arbitrary Sierpi\'{n}ski graph $S_p^n$, and present algorithms to construct the corresponding sequence. These results are obtained by using the main result of the paper, which are sharp bounds for the Grundy domination number of a vertex- and edge-removed graph. That is, given a graph $G$, $e\in E(G)$, and $u\in V(G)$, we prove that $\gamma_{gr}(G)-1\le \gamma_{gr}(G-e) \le \gamma_{gr}(G)+1$ and $\gamma_{gr}(G)-2\le \gamma_{gr}(G-u) \le \gamma_{gr}(G)$. For each of the bounds there exist graphs, in which all three possibilities occur for different edges, respectively vertices.