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.

翻译：我们研究了图的无弦路径凸性（即单调凸性）中的半空间分离问题。在该问题中，给定一个图及两个（不相交的）顶点子集，需要判断这两个集合能否被互补的凸集（称为半空间）所分离。尽管已知该问题在测地凸性（最短路径凸性）中是 $\mathbf{NP}$ 完全的，但我们证明在单调凸性中该问题可在多项式时间内求解。