Computing the hull and interval numbers in the weakly toll convexity

A walk u0u1…uk−1uk of a graph G is a weakly toll walk if u0uk∉E(G), u0ui∈E(G) implies ui=u1, and ujuk∈E(G) implies uj=uk−1. The weakly toll interval of a set S⊆V(G), denoted by I(S), is formed by S and the vertices belonging to some weakly toll walk between two vertices of S. Set S is weakly toll convex if I(S)=S. The weakly toll convex hull of S, denote by H(S), is the minimum weakly toll convex set containing S. The weakly toll interval number of G is the minimum cardinality of a set S⊆V(G) such that I(S)=V(G); and the weakly toll hull number of G is the minimum cardinality of a set S⊆V(G) such that H(S)=V(G). In this work, we show how to compute the weakly toll interval and the weakly toll hull numbers of a graph in polynomial time. In contrast, we show that determining the weakly toll convexity number of a graph G (the size of a maximum weakly toll convex set different from V(G)) is NP-hard.