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arXiv · 2605.10962

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

Ali Zafari·Saeid Alikhani
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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 $T_{2n}(W)$.

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Ali Zafari, Saeid Alikhani. 2026-05-07. The partition dimension and $k$-domination number of a family of non-distance regular graph. https://arxiv.org/abs/2605.10962

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