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

Abstract: This paper investigates the domination relationships among various types of walks connecting two non-adjacent vertices in a graph. In particular, we center our attention on the problem which is proposed in [S. B. Tondato, Graphs Combin. 40 (2024)]. A \textit{( uv )-( m_3 ) path} is a ( uv )-induced path of length at least three. A walk between two non-adjacent vertices in a graph $G$ is called a weakly toll walk if the first and last vertices in the walk are adjacent only to the second and second-to-last vertices, respectively, and these intermediate vertices may appear more than once in the walk. And an $l_k$-path is an induced path of length at most $k$ between two non-adjacent vertices in a graph $G$. We study the domination between weakly toll walks, $l_k$-paths ($k\in \left{2,3\right}$) and different types of walks connecting two non-adjacent vertices $u$ and $v$ of a graph (shortest paths, tolled walks, weakly toll walks, $l_k$-paths for $k\in \left{2,3\right}$, $m_3$-path), and show how these give rise to characterizations of graph classes.