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. © 2013 Springer-Verlag.
CITATION STYLE
Olsen, M., & Revsbæk, M. (2013). Alliances and bisection width for planar graphs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7748 LNCS, pp. 206–216). https://doi.org/10.1007/978-3-642-36065-7_20
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