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A graph is Y Y-reducible if it can be reduced to a vertex by a sequence of series-parallel reductions and Y Y-transformations. Terminals are distinguished vertices which cannot be deleted by reductions and transformations. In this paper we show that four-terminal planar graphs are Y Y-reducible when at least three of the vertices lie on the same face. Using… (More)

We show that for each integer g ≥ 0 there is a constant c g > 0 such that every graph that embeds in the projective plane with sufficiently large face–width r has crossing number at least c g r 2 in the orientable surface Σ g of genus g. As a corollary, we give a polynomial time constant factor approximation algorithm for the crossing number of projective… (More)

For a bipartite graph G we are able to characterize the complete intersection property of the edge subring in terms of the multiplicity and we give optimal bounds for this number. We give a method to obtain a regular sequence for the atomic ideal of G, when G is embedded on an orientable surface. We also give a graph theoretical condition for the edge… (More)

A graph is terminal ∆ − Y-reducible if, it can be reduced to a distinguished set of terminal vertices by a sequence of series-parallel reductions and ∆ − Y-transformations. Terminal vertices (o terminals for short) cannot be deleted by reductions and transformations. Reducibility of terminal graphs is very difficult and in general not possible for graphs… (More)

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs… (More)

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