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

TL;DR

This paper computes the partition dimension and $k$-domination number for Toeplitz graphs, a class of non-distance-regular graphs, filling a gap in their known resolving parameters.

Contribution

It introduces the first explicit calculations of these parameters for Toeplitz graphs, which were previously unknown.

Findings

Partition dimension of Toeplitz graphs determined.

$k$-domination number of Toeplitz graphs computed.

Results provide new insights into resolving parameters of non-distance-regular graphs.

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 $T_{2n}(W)$.