Half-integral flows in a planar graph with four holes

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Suppose that G = (VG, EG) is a planar graph embedded in the euclidean plane, that I, J, K, O are four of its faces (called holes in G), that s1, ..., sr, t1, ..., tr are vertices of G such that each pair {si, ti} belongs to the boundary of some of I,J,K,O, and that the graph (VG, EG∪ {{s1, t1}, ..., {sr, tr}}) is eulerian. We prove that if the multi(commodity)flow problem in G with unit demands on the values of flows from si to ti (i = 1, ..., r) has a solution then it has a half-integral solution as well. In other words, there exist paths P11, P21, P12, P22, ..., p1r, P2r in G such that each pji connects si and ti, and each edge of G is covered at most twice by these paths. (It is known that in case of at most three holes there exist edge-disjoint paths connecting si and ti, i = 1, ..., r, provided that the corres-ponding multiflow problem has a solution, but this is, in general, false in case of four holes.). © 1995.




Karzanov, A. V. (1995). Half-integral flows in a planar graph with four holes. Discrete Applied Mathematics, 56(2–3), 267–295. https://doi.org/10.1016/0166-218X(94)00090-Z

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