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 [Formula: see text] increases by [Formula: see text], where [Formula: see text] 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, [Formula: see text], endowed with a weak nested ear decomposition is equal to [Formula: see text] where [Formula: see text] 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.