Abstract
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.
Research supported in part by NSF Grant CCR-9912418.
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References
J. A. Bondy and U.S.R. Murty, Graph theory with applications, North Holland, New York, 1979.
T. H. Cormen, C. E. Leiserson and R. L. Rivest, An introduction to algorithms, McGraw-Hill, New York, 1990
F. Hoffmann and K. Kriegel, A graph-coloring result and its consequences for polygon-guarding problems, Technical Report TR-B-93-08, Inst. f. Informatik, Freie Universität, Berlin, 1993.
F. Hoffmann and K. Kriegel, A graph-coloring result and its consequences for polygon-guarding problems, SIAM J. Discrete Math, Vol 9(2): 210–224, 1996.
N. Jacobson, Lectures in Abstract Algebras, Springer-Verlag, New York, 1975.
R. J. Lipton, J. Rose and R. E. Tarjan, Generalized Nested Dissection, SIAM J. Numer. Anal., 16: 346–358, 1979.
Huaming Zhang and Xin He, A simple linear time algorithm for finding even triangulations of 2-connected bipartite plane graphs, in Proceedings of ESA’02, LNCS 2461, pp. 902–913.
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Zhang, H., He, X. (2003). On Even Triangulations of 2-Connected Embedded Graphs. In: Warnow, T., Zhu, B. (eds) Computing and Combinatorics. COCOON 2003. Lecture Notes in Computer Science, vol 2697. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45071-8_16
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DOI: https://doi.org/10.1007/3-540-45071-8_16
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