Cosigning Crossing Families and Outer-Planar Gadgets

TL;DR

This paper characterizes the existence of special signings called cosignings in crossing families, provides polynomial algorithms for their construction, and applies these concepts to problems in planar digraphs.

Contribution

It introduces necessary and sufficient conditions for cosignings, develops polynomial-time algorithms, and applies these to outer-planar arc coverings and disjoint dijoins in planar digraphs.

Findings

Polynomial-time algorithms for finding cosignings.

Characterization of cosigning existence via necessary and sufficient conditions.

Application to outer-planar arc coverings and disjoint dijoins in planar digraphs.

Abstract

Let $F$ be a crossing family over ground set $V$, that is, for any two sets $U,W\in{F}$ with nonempty intersection and proper union, both sets $U\cap{W},U\cup{W}$ are in $F$. Let $\sigma:V\to \{+,-\}$ be a signing. We call $\sigma$ a "cosigning" if every set includes a positive element and excludes a negative element. It is "$\cap\cup$-closed" if every pairwise nonempty intersection and co-intersection include positive and negative elements, respectively. We characterize the existence of ($\cap\cup$-closed) cosignings $\sigma$ through necessary and sufficient conditions. Our proofs are algorithmic and lead to elegant `forcing' algorithms for finding $\sigma$, reminiscent of the Cameron-Edmonds algorithm for bicoloring balanced hypergraphs. We prove that the algorithms run in polynomial time, and further, the cosigning algorithm can be run in oracle polynomial time through an application of submodular function minimization. Cosigned crossing families arise naturally in digraphs with vertex set $V$ comprised of sources and sinks, where every set in $F$ is "covered" by an incoming arc. Under mild and necessary conditions, we build an outer-planar arc covering of $F$ when the vertices are placed around a circle. These gadgets are then used to find disjoint dijoins in $0,1$-weighted planar digraphs when the weight-$1$ arcs form a connected component that is not necessarily spanning.