Weakly toll convexity and proper interval graphs

A walk u₀u₁ u₊-₁uₖ is a weakly toll walk if u₀uᵢ E (G) implies uᵢ = u₁ and uⱼuₖ E (G) implies uⱼ=u₊-₁. A set S of vertices of G is weakly toll convex if for any two non-adjacent vertices x, y S any vertex in a weakly toll walk between x and y is also in S. The weakly toll convexity is the graph convexity space defined over weakly toll convex sets. Many studies are devoted to determine if a graph equipped with a convexity space is a convex geometry. An extreme vertex is an element x of a convex set S such that the set S\x\ is also convex. A graph convexity space is said to be a convex geometry if it satisfies the Minkowski-Krein-Milman property, which states that every convex set is the convex hull of its extreme vertices. It is known that chordal, Ptolemaic, weakly polarizable, and interval graphs can be characterized as convex geometries with respect to the monophonic, geodesic, m³, and toll convexities, respectively. Other important classes of graphs can also be characterized in this way. In this paper, we prove that a graph is a convex geometry with respect to the weakly toll convexity if and only if it is a proper interval graph. Furthermore, some well-known graph invariants are studied with respect to the weakly toll convexity.