A Graph-Coloring Result and Its Consequences For Polygon-Guarding Problems
SIAM Journal on Discrete Mathematics
Introduction to Algorithms
A Simple Linear Time Algorithm for Finding Even Triangulations of 2-Connected Bipartite Plane Graphs
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Graph Theory With Applications
Graph Theory With Applications
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Recently, Hoffmann and Kriegel proved an important combinatorial theorem [4]: Every 2-connected bipartite plane graph G has a triangulation in which all vertices have even degree (it's called an even triangulation). Combined with a classical Whitney's Theorem, this result implies that every such a graph has a 3-colorable plane triangulation. Using this theorem, Hoffmann and Kriegel significantly improved the upper bounds of several art gallery and prison guard problems. In [7], Zhang and He presented a linear time algorithm which relies on the complicated algebraic proof in [4]. This proof cannot be extended to similar graphs embedded on high genus surfaces. It's not known whether Hoffmann and Kriegel's Theorem is true for such graphs. In this paper, we describe a totally independent and much simpler proof of the above theorem, using only graph-theoretic arguments. Our new proof can be easily extend to show the existence of even triangulations for similar graphs on high genus surfaces. Hence we show that Hoffmann and Kriegel's theorem remains valid for such graphs. Our new proof leads to a very simple linear time algorithm for finding even triangulations.