Uniform Length Dominating Sequence Graphs

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 $N[v_i]⊈{\cup }_{j=1}^{i-1}N[v_j]$ 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 $k\le 3$. In this article we extend their work by giving a complete characterization of all $k$-uniform graphs with $k\ge 4$.