The paper studies the problem of half-space separation in the monophonic convexity of graphs. Given a graph G and two disjoint, convex subsets A and B of the vertices, the goal is to determine whether A and B can be separated by complementary convex half-spaces.
The key insights and steps of the algorithm are:
Linkage along a shortest path: The authors show that A and B are separable if and only if there exists a vertex on a shortest path between a vertex in A and a vertex in B that can be used to link A and B into separable sets.
Saturation with the hull operator: The authors define the saturation of A and B, denoted S(A, B) and S(B, A), which preserves separability. Saturation can be computed in polynomial time.
Testing bipartiteness: The authors characterize the separability of the saturated sets S(A, B) and S(B, A) in terms of the bipartiteness of an associated graph GAB and the absence of certain "forbidden pairs" of vertices. This test can also be performed in polynomial time.
The main result is that half-space separation in monophonic convexity can be decided in polynomial time, in contrast with the NP-completeness of the problem for geodesic convexity.
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