Length Uniformity in Legal Dominating Sequences
Aysel Erey · arXiv (Cornell University) · 2019
A sequence of vertices $(v_1,\, \dots , \,v_k)$ of a graph $G$ is called a {\it legal dominating sequence} if $\{v_1,\, \dots , \,v_k\}$ is a dominating set of $G$ and $N[v_i] subseteq \cup _{j=1}^{i-1} N[v_j]$ for every $i$. A graph $G$ is said to be {\it $k-$uniform} if all legal dominating sequences have equal length $k$. Bre{\u s}ar et al. (2014) characterized $k$-uniform graphs with $k\leq 3$. In this article we extend their work by giving a complete characterization of all $k$-uniform graphs with $k\geq 4$. We also discuss a variant of this problem for another type of sequence where open neighborhoods are considered instead of closed neighborhoods.