Abstract:
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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 l increases by \lfloor{l/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 [(|VG|\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. |