On finite convexity spaces induced by sets of paths in graphs

A finite convexity space is a pair ( V , C ) consisting of a finite set V and a set C of subsets of V such that 0̸ ∈ C , V ∈ C , and C is closed under intersection. A graph G with vertex set V and a set P of paths of G naturally define a convexity space ( V , C ( P ) ) where C ( P ) contains all subsets C of V such that whenever C contains the endvertices of some path P in P , then C contains all vertices of P . ve that for a finite convexity space ( V , C ) and a graph G with vertex set V , there is a set P of paths of G with C = C ( P ) if and only if • set S which is not convex with respect to C contains two distinct vertices whose convex hull with respect to C is not contained in S and ery two elements x and z of V and every element y distinct from x and z of the convex hull of { x , z } with respect to C , the subgraph of G induced by the convex hull of { x , z } with respect to C contains a path between x and z with y as an internal vertex. ermore, we prove that the corresponding algorithmic problem can be solved efficiently.