Abstract
This paper considers the conjecture by Grünbaum that every planar 3-connected graph has a spanning tree T such that both T and its co-tree have maximum degree at most 3. Here, the co-tree of T is the spanning tree of the dual obtained by taking the duals of the non-tree edges. While Grünbaum’s conjecture remains open, we show that every planar 3-connected graph has a spanning tree T such that both T and its co-tree have maximum degree at most 5. It can be found in linear time.
Supported by NSERC and the Ross and Muriel Cheriton Fellowship. Research initiated while participating at Dagstuhl seminar 13421.
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References
Badent, M., Baur, M., Brandes, U., Cornelsen, S.: Leftist canonical ordering. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 159–170. Springer, Heidelberg (2010)
Barnette, D.W.: Trees in polyhedral graphs. Canad. J. Math. 18, 731–736 (1966)
Barnette, D.W.: 3-trees in polyhedral maps. Israel Journal of Mathematics 79, 251–256 (1992)
boost C + + libraries on planar graphs (2013), http://www.boost.org/ (last accessed December 2, 2013)
Chrobak, M., Kant, G.: Convex grid drawings of 3-connected planar graphs. Technical Report RUU-CS-93-45, Rijksuniversiteit Utrecht (1993)
Chrobak, M., Kant, G.: Convex grid drawings of 3-connected planar graphs. Internat. J. Comput. Geom. Appl. 7(3), 211–223 (1997)
de Fraysseix, H., Ossona de Mendez, P.: P.I.G.A.L.E., Public Implementation of Graph Algorithm Libeary and Editor (2013), http://pigale.sourceforge.net/ (last accessed December 2, 2013)
Felsner, S.: Convex drawings of planar graphs and the order dimension of 3-polytopes. Order 18, 19–37 (2001)
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)
Gao, Z., Richter, R.B.: 2-walks in circuit graphs. J. Comb. Theory, Ser. B 62(2), 259–267 (1994)
Grünbaum, B.: Polytopes, graphs, and complexes. Bull. Amer. Math. Soc. 76, 1131–1201 (1970)
Grünbaum, B.: Graphs of polyhedra; polyhedra as graphs. Discrete Mathematics 307(3-5), 445–463 (2007)
He, X., Kao, M.-Y., Lu, H.-I.: Linear-time succinct encodings of planar graphs via canonical orderings. SIAM J. Discrete Math. 12(3), 317–325 (1999)
Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16, 4–32 (1996)
Miura, K., Azuma, M., Nishizeki, T.: Canonical decomposition, realizer, Schnyder labeling and orderly spanning trees of plane graphs. Int. J. Found. Comput. Sci. 16(1), 117–141 (2005)
Nishizeki, T., Rahman, M.S.: Planar Graph Drawing. Lecture Notes Series on Computing, vol. 12. World Scientific (2004)
Ozeki, K., Yamashita, T.: Spanning trees: A survey. Graphs and Combinatorics 27(1), 1–26 (2011)
Strothmann, W.-B.: Bounded-degree spanning trees. PhD thesis, FB Mathematik/Informatik und Heinz-Nixdorf Institute, Universität-Gesamthochschule Paderborn (1997)
Tamassia, R., Tollis, I.: A unified approach to visibility representations of planar graphs. Discrete Computational Geometry 1, 321–341 (1986)
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Biedl, T. (2014). Trees and Co-trees with Bounded Degrees in Planar 3-connected Graphs. In: Ravi, R., Gørtz, I.L. (eds) Algorithm Theory – SWAT 2014. SWAT 2014. Lecture Notes in Computer Science, vol 8503. Springer, Cham. https://doi.org/10.1007/978-3-319-08404-6_6
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DOI: https://doi.org/10.1007/978-3-319-08404-6_6
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