A sequence of vertices $(v_1,\, \dots , \,v_k)$ of a graph $G$ is called a { \it dominating closed neighborhood sequence} if $\{ v_1,\, \dots , \,v_k\}$ is a dominating set of $G$ and $N[v_i]\nsubseteq \cup _{ j=1}^{ i-1} N[v_j]$ for every $i$. A graph $G$ is said to be { \it $k-$uniform} if all dominating closed neighborhood sequences have equal length $k$. Bre{ \v 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 $k\geq 4$.
Categories: math.CO