Tolerant Radon partitions on the all-paths convexity in graphs

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