On Walk Domination: Weakly Toll Domination, l2 and l3 Domination

Abstract In this paper we study domination between different types of walks connecting two non-adjacent vertices of a graph. In particular, we center our attention on weakly toll walk and lk-path for k ∈ {2, 3}. A walk between two non-adjacent vertices in a graph G is called a weakly toll walk if the first and the last vertices in the walk are adjacent, respectively, only to the second and second-to-last vertices, which may occur more than once in the walk. And an lk-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, lk-paths (k ∈ {2, 3}) and different types of walks connecting two non-adjacent vertices u and v of a graph (shortest paths, induced paths, paths, tolled walks, weakly toll walks, lk-paths for k ∈ {3, 4}), and show how these give rise to characterizations of graph classes.