Tolerant Radon partitions on the all-paths convexity in graphs

Abstract

In a graph G, the all-paths transit function A(u, v), consists of the set of all vertices in the graph G which lies in some path connecting u and v. Convexity obtained by the all-paths transit function is called all-paths convexity. A partition of a set P of vertices of a graph G into two disjoint non-empty subsets is called a Radon partition if the convex hulls of the subsets intersect. A Radon partition (P0, Q0) of P into nonempty subsets is called t-tolerant Radon partition, if for any set S ⊆ P with | S | ≤ t , the intersection of the convex hulls 〈 P 0 − S 〉 ∩ 〈 Q 0 − S 〉 ≠ ∅ . Here we study the t-tolerant Radon number of G with respect to all-paths convexity. It is the minimum number k such that any collection of k vertices of G has t-tolerant Radon partition. We prove that if G has only one block, then k = 2 t + 2 whenever t ≥ 1 . Finally, we prove that if the block-cut vertex tree representation of G is a path then k = 2 t + 3 , and if the block-cut vertex tree representation of G is a tree that is not a path then k = 2 t + 4 .