Monotonicity of total Betti numbers under contracting the connecting edge

Ascertain whether, for every graph G obtained by connecting two disjoint connected graphs G1 and G2 by a single edge e = {x,y}, the inequality βi(K[G]) ≥ βi(K[G e]) holds for all i ≥ 0.

Background

The paper shows that for even path contractions, total Betti numbers weakly decrease (Theorem 2.7), and in certain cases equality holds (Theorem 2.9). For edge contractions in graphs connected by an edge, the behavior is more delicate: examples exhibit both preservation and decrease, and Theorem 4.2 only guarantees β1 invariance.

This question seeks a general monotonicity principle for total Betti numbers under contracting the connecting edge in this special class of graphs.