Half-space separation in monophonic convexity

TL;DR

This paper investigates the problem of separating two vertex sets with convex sets in monophonic convexity, showing it is polynomial-time solvable, unlike the NP-complete case in geodesic convexity.

Contribution

It demonstrates that half-space separation in monophonic convexity can be decided efficiently, contrasting with the complexity in geodesic convexity.

Findings

Half-space separation is polynomial-time solvable in monophonic convexity.

NP-completeness holds for geodesic convexity, but not for monophonic convexity.

The study advances understanding of convexity-based graph separation problems.

Abstract

We study half-space separation in the convexity of chordless paths of a graph, i.e., monophonic convexity. In this problem, one is given a graph and two (disjoint) subsets of vertices and asks whether these two sets can be separated by complementary convex sets, called half-spaces. While it is known this problem is $\mathbf{NP}$-complete for geodesic convexity -- the convexity of shortest paths -- we show that it can be solved in polynomial time for monophonic convexity.