Flipping odd matchings in geometric and combinatorial settings

2025-08-25

AI Summary

This paper investigates the reconfiguration of odd matchings—maximum matchings covering all but one vertex—on combinatorial graphs and geometric point sets, where the reconfiguration operation is a “flip”: swapping the matching status of the unmatched vertex with one endpoint of a matched edge. Methodologically, it integrates graph theory, computational geometry, parameterized algorithms, and order-type analysis. The contributions include: (i) the first polynomial-time verifiable necessary and sufficient characterization of odd-matching reconfigurability in the combinatorial setting; (ii) an FPT algorithm for computing flip distance on convex point sets, along with a general framework reducing order types to general position; (iii) theoretical results showing that the combinatorial flip graph has linear diameter, and that deciding the existence of a flip sequence of length ≤ k is NP-complete in both settings—but FPT on convex point sets.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We study the problem of reconfiguring odd matchings, that is, matchings that cover all but a single vertex. Our reconfiguration operation is a so-called flip where the unmatched vertex of the first matching gets matched, while consequently another vertex becomes unmatched. We consider two distinct settings: the geometric setting, in which the vertices are points embedded in the plane and all occurring odd matchings are crossing-free, and a combinatorial setting, in which we consider odd matchings in general graphs. For the latter setting, we provide a complete polynomial time checkable characterization of graphs in which any two odd matchings can be reconfigured into each another. This complements the previously known result that the flip graph is always connected in the geometric setting [Aichholzer, Brötzner, Perz, and Schnider. Flips in odd matchings]. In the combinatorial setting, we prove that the diameter of the flip graph, if connected, is linear in the number of vertices. Furthermore, we establish that deciding whether there exists a flip sequence of length $k$ transforming one given matching into another is NP-complete in both the combinatorial and the geometric settings. To prove the latter, we introduce a framework that allows us to transform partial order types into general position with only polynomial overhead. Finally, we demonstrate that when parameterized by the flip distance $k$, the problem is fixed-parameter tractable (FPT) in the geometric setting when restricted to convex point sets.

Problem

• Characterizing reconfigurability of odd matchings in general graphs via flips
• Determining computational complexity of flip sequence problems in both settings
• Analyzing flip graph connectivity and diameter properties for odd matchings

Innovation

• Using flip operations to reconfigure odd matchings
• Developing polynomial framework for order types
• Establishing FPT algorithm for convex sets