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

Given two non-adjacent vertices $u$ and $v$, we say a $uv$-walk $W$ dominates a $uv$-walk $W^{\mathrm{\prime }}$ if every internal vertex of $W^{\mathrm{\prime }}$ 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 the problem which is proposed in [S. B. Tondato, Graphs Combin. 40 (2024)]. We study the domination between different walk types (shortest paths, toll walks, weakly toll walks, $l_k$-paths for $k\in \{2,3\}$) and $m_3$-paths. And we show how these relationships give rise to characterizations of graph classes.