On Alliance Partitions and Bisection Width for Planar Graphs

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 how to construct infinitely many non-trivial examples of graphs that can not be partitioned into alliances and 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. Submitted: May 2013 Reviewed: September 2013 Revised: October 2013 Accepted: October 2013 Final: October 2013 Published: November 2013 Article type: Regular paper Communicated by: S. K. Ghosh E-mail addresses: [email protected] (Martin Olsen) [email protected] (Morten Revsbæk) ∗Center for Massive Data Algorithmics, a center of the Danish National Research Foundation. 600 Olsen and Revsbæk On Alliance Partitions and Bisection Width for Planar Graphs