Minimum blockers and extremal graphs for nonnested matchings

Abstract

We classify all smallest sets of edge deletions that destroy every nonnested perfect matching in a complete ordered graph. The vertices have a fixed linear order. A perfect matching selects edges that use each vertex exactly once; it is nonnested when no selected edge has both endpoints strictly between those of another. On $2k$ vertices, exactly $k$ deletions are necessary, and we describe all $2^k+k-2$ minimum deletion sets for every $k\ge 2$. Allowing a matching to leave vertices unused changes the extremal problem. Barát, Freschi and Tóth proposed that an $n$-vertex ordered graph avoiding a nonnested $k$-edge matching can have at most $(k-1)n$ edges. We construct counterexamples for every $k\ge 5$ and $n\ge 2k+1$. For $n\ge 3k$, our constructions exceed the proposed value by a number of edges proportional to $k^2$, matching the order of the known upper bound on this excess. The classification follows from cuts between consecutive intervals; the larger constructions coordinate deletions at the two ends of the vertex order. The exact extremal value remains open in general.

arXiv comment30 pages, 11 figures, 1 algorithm. Substantially expanded version: new constructions with quadratic excess beyond the proposed extremal bound; expanded proofs and illustrations