Abstract
An alliance in a graph is a set of vertices (allies) such that each vertex in the alliance has at least as many allies (counting the vertex itself) as non-allies in its neighborhood of the graph. We show that any planar graph with minimum degree at least 4 can be split into two alliances in polynomial time. We base this on a proof of an upper bound of n on the bisection width for 4-connected planar graphs with an odd number of vertices. This improves a recently published n + 1 upper bound on the bisection width of planar graphs without separating triangles and supports the folklore conjecture that a general upper bound of n exists for the bisection width of planar graphs.
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Olsen, M., Revsbæk, M. (2013). Alliances and Bisection Width for Planar Graphs. In: Ghosh, S.K., Tokuyama, T. (eds) WALCOM: Algorithms and Computation. WALCOM 2013. Lecture Notes in Computer Science, vol 7748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36065-7_20
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DOI: https://doi.org/10.1007/978-3-642-36065-7_20
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