The partition dimension and $k$-domination number of a family of non-distance regular graph

Abstract

A partition $\Sigma =\{S_1,S_2,\dots ,S_k\}$ of the vertex set $V(G)$ is a resolving partition if every pair of distinct vertices in $G$ has a unique representation relative to $\Sigma$. The partition dimension, $pd(G)$, is the minimum cardinality of such a partition. Additionally, a subset $D\subseteq V(G)$ is a $k$-dominating set if every vertex in $V(G)\setminus D$ has at least $k$ neighbors in $D$; the $k$-domination number, ${\gamma }_k(G)$, denotes the minimum size of such a set. Determining these parameters is NP-complete and particularly challenging for non-distance-regular graphs. This paper consider the Toeplitz graph $T_{2n}(W)$, a family of non-distance-regular graphs. While some resolving parameters for this family have been established, its partition dimension and $k$-domination number remain unknown. We close this gap by computing both parameters for $T_{2n}(W)$.

11 pages