Abstract
A tree-partition of a graph is a partition of its vertices into ‘bags’ such that contracting each bag into a single vertex gives a forest. It is proved that every k-tree has a tree-partition such that each bag induces a (k-1)-tree, amongst other properties. Applications of this result to two well-studied models of graph layout are presented. First it is proved that graphs of bounded tree-width have bounded queue-number, thus resolving an open problem due to Ganley and Heath [2001] and disproving a conjecture of Pemmaraju [1992]. This result provides renewed hope for the positive resolution of a number of open problems regarding queue layouts. In a related result, it is proved that graphs of bounded tree-width have three-dimensional straight-line grid drawings with linear volume, which represents the largest known class of graphs with such drawings.
Research supported by NSERC and FCAR.
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References
Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math. 23(1), 11–24 (1989)
Cohen, R.F., Eades, P., Lin, T., Ruskey, F.: Three-dimensional graph drawing. Algorithmica 17(2), 199–208 (1996)
Di Giacomo, E., Liotta, G., Wismath, S.: Drawing series-parallel graphs on a box. In: Proc. 14th Canadian Conf. on Computational Geometry (CCCG 2002), pp. 149–153. The Univ. of Lethbridge, Canada (2002)
Ding, G., Oporowski, B.: Some results on tree decomposition of graphs. J. Graph Theory 20(4), 481–499 (1995)
Dujmović, V., Morin, P., Wood, D.R.: Path-width and three-dimensional straight-line grid drawings of graphs. In: Advanced Symbolic Analysis for Compilers. LNCS, vol. 2628, pp. 42–53. Springer, Heidelberg (2002)
Dujmović, V., Wood, D.R.: Tree-partitions of k-trees with applications in graph layout. Technical Report TR-02-03, School of Computer Science, Carleton Univ., Ottawa, Canada (2002)
Dujmović, V., Wood, D.R.: Three-Dimensional Grid Drawings with Sub- Quadratic Volume. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 190–201. Springer, Heidelberg (2004)
Felsner, S., Liotta, G., Wismath, S.: Straight-line drawings on restricted integer grids in two and three dimensions. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 328–342. Springer, Heidelberg (2002)
Ganley, J.L., Heath, L.S.: The pagenumber of k-trees is O(k). Discrete Appl. Math. 109(3), 215–221 (2001)
Gyárfás, A., West, D.: Multitrack interval graphs. In: 26th Southeastern Conf. on Combinat, Graph Theory and Comput., Congr. Numer, vol. 109, pp. 109–116 (1995)
Heath, L.S., Leighton, F.T., Rosenberg, A.L.: Comparing queues and stacks as mechanisms for laying out graphs. SIAM J. Disc. Math. 5, 398–412 (1992)
Heath, L.S., Rosenberg, A.L.: Laying out graphs using queues. SIAM J. Comput. 21(5), 927–958 (1992)
Lin, Y., Li, X.: Pagenumber and treewidth. Disc. Applied Math. (to appear)
Pach, J., Thiele, T., Tóth, G.: Three-dimensional grid drawings of graphs. In: DiBattista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 47–51. Springer, Heidelberg (1997); Also in Advances in discrete and computational geometry, Contemporary Mathematics 223, 251–255, Amer. Math. Soc. (1999)
Pemmaraju, S.V.: Exploring the Powers of Stacks and Queues via Graph Layouts. PhD thesis, Virginia Polytechnic Institute and State Univ., Virginia, U.S.A (1992)
Reed, B.A.: Algorithmic aspects of tree width. In: Recent advances in algorithms and combinatorics, pp. 85–107. Springer, Heidelberg (2003)
Rengarajan, S., Veni Madhavan, C.E.: Stack and queue number of 2- trees. In: Li, M., Du, D.-Z. (eds.) COCOON 1995. LNCS, vol. 959, pp. 203–212. Springer, Heidelberg (1995)
Rose, D.J., Tarjan, R.E., Leuker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5, 266–283 (1976)
Seese, D.: Tree-partite graphs and the complexity of algorithms. In: Budach, L. (ed.) FCT 1985. LNCS, vol. 199, pp. 412–421. Springer, Heidelberg (1985)
Wood, D.R.: Queue layouts, tree-width, and three-dimensional graph drawing. In: Agrawal, M., Seth, A.K. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 348–359. Springer, Heidelberg (2002)
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Dujmović, V., Wood, D.R. (2003). Tree-Partitions of k-Trees with Applications in Graph Layout. In: Bodlaender, H.L. (eds) Graph-Theoretic Concepts in Computer Science. WG 2003. Lecture Notes in Computer Science, vol 2880. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39890-5_18
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