Tolerant Radon partitions of induced path convexity in graphs

In a graph [Formula: see text], the induced path transit function [Formula: see text] consists of the set of all vertices in [Formula: see text] that lie on some induced path connecting the vertices [Formula: see text] and [Formula: see text]. The convexity derived from this transit function is referred to as induced path convexity. A partition of a set [Formula: see text] into two nonempty, disjoint subsets is called a Radon partition if the convex hulls of the subsets intersect. A Radon partition [Formula: see text] is said to be a [Formula: see text]-tolerant Radon partition if, for any subset [Formula: see text] with [Formula: see text], the intersection of the convex hulls [Formula: see text]. This paper investigates the [Formula: see text]-tolerant Radon partition of graphs with respect to induced path convexity. For any noncomplete graph [Formula: see text] without clique separators, we show that for any subset [Formula: see text] with [Formula: see text], there exists a [Formula: see text]-tolerant Radon partition for [Formula: see text]. We also prove the existence of [Formula: see text]-tolerant Radon partitions of the induced path convexity in block graphs.