Core Concepts

The paper presents a polynomial-time algorithm to decide whether two disjoint, convex sets in a graph can be separated by complementary convex half-spaces in the monophonic convexity.

Abstract

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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Key Insights Distilled From

by Mohammed Ela... at **arxiv.org** 04-29-2024

Deeper Inquiries

The complexity of the half-space separation problem varies across different graph convexities. In the context of the paper, the problem was studied for monophonic convexity, where it was shown to be solvable in polynomial time. This is in contrast to geodesic convexity, where the problem is known to be NP-complete.
For other graph convexities beyond monophonic and geodesic convexity, the complexity of the half-space separation problem may differ. The paper mentions various graph convexities like m3-convexity, triangle-path convexity, toll convexity, and weakly toll convexity, but the complexity status of the half-space separation problem for these convexities is not explicitly addressed.
Further research would be needed to determine the complexity of the half-space separation problem for these other graph convexities and to compare it with the complexities observed for monophonic and geodesic convexity.

Efficient separation of monophonic convex sets in graphs can be particularly useful in various real-world applications. One such application could be in network security, where identifying and separating different components or clusters within a network based on their connectivity patterns can help in detecting and preventing security breaches.
Another application could be in transportation and logistics, where separating regions of a transportation network based on their connectivity can optimize route planning, traffic management, and resource allocation.
Additionally, in social network analysis, separating groups of individuals based on their interactions and connections can provide insights into community detection, influence propagation, and targeted marketing strategies.

The techniques developed in the paper for solving the half-space separation problem in monophonic convexity could potentially be extended to solve other problems related to graph convexity, such as the p-partition problem.
The p-partition problem involves partitioning the vertices of a graph into p convex sets based on the specific convexity at hand. By adapting the algorithm and concepts used for half-space separation in monophonic convexity, it may be possible to develop efficient algorithms for solving the p-partition problem for different graph convexities.
Further research and analysis would be required to explore the applicability and effectiveness of extending these techniques to solve other graph convexity problems like the p-partition problem.

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