Conditions for preservation of total Betti numbers under contraction of the connecting edge

Determine necessary and sufficient conditions under which contracting the connecting edge e = {x,y} in a graph G that is formed by joining two disjoint connected graphs G1 and G2 by e preserves all total Betti numbers of the edge ring K[G], i.e., βi(K[G]) = βi(K[G e]) for every i ≥ 0.

Background

Section 4 introduces graphs "connected by the edge e" (G = G1 −e− G2) and studies how contracting e impacts the Betti numbers of K[G]. Theorem 4.2 shows β1 is preserved, and Proposition 4.3 gives full preservation when one component is bipartite. However, examples demonstrate that behavior can vary depending on how the components are connected by e, and equality may or may not hold.

This motivates the general problem of identifying precise structural conditions on G1, G2, and the choice of vertices x ∈ V(G1), y ∈ V(G2) that guarantee preservation of all total Betti numbers under contracting e.