We show that the maximum flow problem can be solved in near-linear time for K5-minor-free and K3,3-minor-free graphs. The same result also holds for H-minor-free graphs when H can be embedded in the plane with one crossing and a structural decomposition of the input flow graph is given.
Tutte's method of spring embeddings allows any triangulated planar graph to be drawn so that the outer face has any pre-specified convex shape, but it may place vertices exponentially close to each other. Alternative graph drawing methods provide polynomial-area straight line drawings but do not allow the outer face shape to be specified. We describe a drawing method that combines both properties: it has polynomial area, and can match any pre-specified shape of the outer face, even a shape in which some of the vertices have 180 degree angles. We apply our results to drawing polygonal schemas for graphs embedded on surfaces of positive genus.
We lower bound the height of a drawing of a planar graph in an integer grid by a topological invariant, the homotopy height, and show that this invariant is equal to the smallest height of a grid graph in which the given graph is a minor.
We describe a random distribution on the spanning trees of bounded-bandwidth graphs such that each edge has bounded expected stretch, along with several related results for other kinds of graph widths.
Co-authors – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine
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