On walk domination: Between different types of walks and $m_3$-path

Hangdi Chen; Yuhan Ma; Qingjie Ye

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arXiv · 2504.05953

On walk domination: Between different types of walks and $m_3$-path

Hangdi Chen·Yuhan Ma·Qingjie Ye

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Abstract

Given two non-adjacent vertices $ u $ and $ v $, we say a $uv$-walk $ W $ dominates a $uv$-walk $ W' $ if every internal vertex of $ W' $ is adjacent to some internal vertex of $ W $ or belongs to $ W $. A class of walks $\mathbf{A}$ dominates a class of walks $\mathbf{B}$ if for every pair of non-adjacent vertices $u,v$ in the graph, every $uv$-walk in $\mathbf{A}$ dominates every $uv$-walk in $\mathbf{B}$. This paper investigates the domination relationships among various types of walks connecting two non-adjacent vertices in a graph. In particular, we focus on a problem proposed by Tondato (2024). We study the domination between different walk types (shortest paths, toll walks, weakly toll walks, $l_k$-paths for $k\in \left\{2,3\right\}$) and $m_3$-paths. Furthermore, we show how these relationships give rise to characterizations of graph classes.

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Hangdi Chen, Yuhan Ma, Qingjie Ye. 2025-04-08. On walk domination: Between different types of walks and $m_3$-path. https://arxiv.org/abs/2504.05953

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