Abstract. We show that for pairs [Formula: see text] and [Formula: see text] of disjoint subsets of vertices of a graph [Formula: see text], if [Formula: see text] is sufficiently large, then there exists a vertex [Formula: see text] in [Formula: see text] such that there are two ways to reduce [Formula: see text] by a vertex-minor operation that removes [Formula: see text] while preserving the connectivity between [Formula: see text] and [Formula: see text] and the connectivity between [Formula: see text] and [Formula: see text]. Our theorem implies an analogous theorem of Chen and Whittle (SIAM J. Discrete Math., 28 (2014), pp. 1402–1404) for matroids restricted to binary matroids.

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