Regularity of the vanishing ideal over a bipartite nested ear decomposition

We study the Castelnuovo-Mumford regularity of the vanishing ideal over a bipartite graph endowed with a decomposition of its edge set. We prove that, under certain conditions, the regularity of the vanishing ideal over a bipartite graph obtained from a graph by attaching a path of length $\ell$ increases by $\lfloor \frac{\ell}{2}\rfloor (q-2)$, where $q$ is the order of the field of coefficients. We use this result to show that the regularity of the vanishing ideal over a bipartite graph, $G$, endowed with a weak nested ear decomposition is equal to $$\textstyle \frac{|V_G|+ \epsilon -3}{2}(q-2),$$ where $\epsilon$ is the number of even length ears and pendant edges of the decomposition. As a corollary, we show that for bipartite graph, the number of even length ears in a nested ear decomposition starting from a vertex is constant.