TL;DR
This paper investigates the structure of graphs derived from stable matching instances, proving recognition complexity results, developing algorithms, and characterizing these graphs within certain classes.
Contribution
It introduces complexity results, algorithms, and characterizations for graphs of stably matchable pairs, advancing understanding of their structural properties.
Findings
Recognition of these graphs is NP-complete.
An exponential-time recognition algorithm based on edges.
A linear-vertex, exponential-in-polynomial carving width algorithm.
Abstract
We study the graphs formed from instances of the stable matching problem by connecting pairs of elements with an edge when there exists a stable matching in which they are matched. Our results include the NP-completeness of recognizing these graphs, an exact recognition algorithm that is singly exponential in the number of edges of the given graph, and an algorithm whose time is linear in the number of vertices of the graph but exponential in a polynomial of its carving width. We also provide characterizations of graphs of stably matchable pairs that belong to certain classes of graphs, and of the lattices of stable matchings that can have graphs in these classes.
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