Art gallery theorems and algorithms
Art gallery theorems and algorithms
Introduction to algorithms
ACM SIGACT News
A Graph-Coloring Result and Its Consequences For Polygon-Guarding Problems
SIAM Journal on Discrete Mathematics
Graph Theory With Applications
Graph Theory With Applications
On even triangulations of 2-connected embedded graphs
COCOON'03 Proceedings of the 9th annual international conference on Computing and combinatorics
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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 result, Hoffmann and Kriegel significantly improved the upper bounds of several art gallery and prison guard problems. A complicated O(n2) time algorithm was obtained in [4] for constructing an even triangulation of G. Hoffmann and Kriegel conjectured that there is an O(n3/2) algorithm for solving this problem [4].In this paper, we develop a very simple O(n) time algorithm for solving this problem. Our algorithm is based on thorough study of the structure of all even triangulations of G. We also obtain a simple formula for computing the number of distinct even triangulations of G.