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

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.