$\Gamma$-graphic delta-matroids and their applications

TL;DR

This paper introduces $ ext{Gamma}$-graphic delta-matroids derived from $ ext{Gamma}$-labelled graphs, providing polynomial algorithms for key problems and exploring their properties and applications in graph theory.

Contribution

It defines $ ext{Gamma}$-graphic delta-matroids, proves their properties, and offers polynomial algorithms for the separation problem and maximum weight packings, extending delta-matroid theory.

Findings

A collection of edge sets forms a delta-matroid called $ ext{Gamma}$-graphic delta-matroid.

A polynomial-time algorithm solves the separation problem for these delta-matroids.

Applications include algorithms for maximum weight packings of trees and $S$-tree packings.

Abstract

For an abelian group $\Gamma$, a $\Gamma$-labelled graph is a graph whose vertices are labelled by elements of $\Gamma$. We prove that a certain collection of edge sets of a $\Gamma$-labelled graph forms a delta-matroid, which we call a $\Gamma$-graphic delta-matroid, and provide a polynomial-time algorithm to solve the separation problem, which allows us to apply the symmetric greedy algorithm of Bouchet to find a maximum weight feasible set in such a delta-matroid. We present two algorithmic applications on graphs; Maximum Weight Packing of Trees of Order Not Divisible by $k$ and Maximum Weight $S$-Tree Packing. We also discuss various properties of $\Gamma$-graphic delta-matroids.