$\mathrm{\Gamma }$-graphic delta-matroids and their applications

For an abelian group $\mathrm{\Gamma }$, a $\mathrm{\Gamma }$-labelled graph is a graph whose vertices are labelled by elements of $\mathrm{\Gamma }$. We prove that a certain collection of edge sets of a $\mathrm{\Gamma }$-labelled graph forms a delta-matroid, which we call a $\mathrm{\Gamma }$-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 $\mathrm{\Gamma }$-graphic delta-matroids.