Abstract
In this paper, we study a layout problem of a digraph using queues. The queuenumber of a digraph is the minimum number of queues required for a queue layout of the digraph. We present upper and lower bounds on the queuenumber of an iterated line digraph L k(G) of a digraph G. In particular, our upper bound depends only on G and is independent of the number of iterations k. Queue layouts can be applied to three-dimensional drawings. From the result on the queuenumber of L k(G), it is shown that for any fixed digraph G, L k(G) has a three-dimensional drawing with O(n) volume, where n is the number of vertices in L k(G). We also apply these results to particular families of iterated line digraphs such as de Bruijn digraphs, Kautz digraphs, butterfly digraphs, and wrapped butterfly digraphs.
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Alon, N., McDiarmid, C., Reed, B.: Acyclic coloring of graphs. Random Structures & Algorithms 2, 277–288 (1991)
Bermond, J.-C., Darrot, E., Delmas, O., Perennes, S.: Hamiltonian circuits in the directed wrapped butterfly network. Discrete Applied Math. 84, 21–42 (1998)
Bermond, J.-C., Peyrat, C.: De Bruijn and Kautz networks: A competition for the hypercube? In: André, F., Verjus, J.P. (eds.) Hypercube and distributed computers, pp. 279–293. North Holland, Amsterdam (1989)
Bernhart, F., Kainen, P.C.: The book thickness of a graph. J. Combin. Theory Ser. B 27, 320–331 (1979)
Chung, F.R.K., Leighton, F.T., Rosenberg, A.L.: Embedding graphs in books: a layout problem with application to VLSI design. SIAM J. Algebraic Discrete Methods 8, 33–58 (1987)
Dujmović, V., Wood, D.R.: Tree-Partitions of k-trees with applications in graph layouts. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 205–217. Springer, Heidelberg (2003)
Dujmović, V., Wood, D.R.: New results in graph layout, Tech. Report TR-2003-04, School of Computer Science, Carleton University, Canada (2002)
Dujmović, V., Morin, P., Wood, D.R.: Path-width and three-dimensional straight-line grid drawings of graphs. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 42–53. Springer, Heidelberg (2002)
Even, S., Itai, A.: Queues, stacks and graphs. In: Kohavi, Z., Paz, A. (eds.) Theory of Machines and Computations, pp. 71–86. Academic Press, New York (1971)
Fertin, G., Raspaud, A., Reed, B.: On star coloring of graphs. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 140–153. Springer, Heidelberg (2001)
Fiol, M.A., Yebra, J.L.A., Alegre, I.: Line digraph iterations and the (d,k) digraph problem. IEEE Trans. Comput. 33, 400–403 (1984)
Games, R.A.: Optimal book embeddings of the FFT, Benes, and barrel shifter networks. Algorithmica 1, 233–250 (1986)
Ganley, J.L., Heath, L.S.: The pagenumber of k-trees is O(k). Discrete Applied Math. 109, 215–221 (2001)
Hasunuma, T.: Embedding iterated line digraphs in books. Networks 40, 51–62 (2002)
Hasunuma, T.: Queuenumbers and stacknumbers of generalized de Bruijn and Kautz digraphs (submitted)
Hasunuma, T., Shibata, Y.: Embedding de Bruijn, Kautz and shuffle-exchange networks in books. Discrete Applied Math. 78, 103–116 (1997)
Hasunuma, T., Shibata, Y.: Containment of butterflies in networks constructed by the line digraph operation. Inform. Process. Lett. 61, 25–30 (1997)
Heath, L.S., Leighton, F.T., Rosenberg, A.L.: Comparing queues and stacks as mechanisms for laying out graphs. SIAM J. Discrete Math. 5 (1992)
Heath, L.S., Pemmaraju, S.V., Trenk, A.N.: Stack and queue layouts of directed acyclic graphs I. SIAM J. Comput. 28, 1510–1539 (1999)
Heath, L.S., Pemmaraju, S.V.: Stack and queue layouts of directed acyclic graphs II. SIAM J. Comput. 28, 1588–1626 (1999)
Heath, L.S., Rosenberg, A.L.: Laying out graphs using queues. SIAM J. Comput. 21, 927–958 (1992)
Konoe, M., Hagiwara, K., Tokura, N.: On the pagenumber of hypercubes and cube-connected cycles. IEICE Trans. J71-D, 490–500 (1988) (in Japanese)
Malitz, S.M.: Genus g graphs have pagenumber O(\(\sqrt{g}\)). J. Algorithms 17, 85–109 (1994)
Muder, D.J., Weaver, M.L., West, D.B.: Pagenumber of complete bipartite graphs. J. Graph Theory 12, 469–489 (1988)
Pemmaraju, S.V.: Exploring the powers of stacks and queues via graph layouts, Ph.D thesis, Virginia Polytechnic Institute and State University, Virginia, USA (1992)
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)
Rosenberg, A.L.: The Diogenes approach to testable fault-tolerant arrays of processors. IEEE Trans. Comput. C-32, 902–910 (1983)
Swaminathan, R.P., Giriaj, D., Bhatia, D.K.: The pagenumber of the class of bandwidth-k graphs is k – 1. Inform. Process. Lett. 55, 71–74 (1995)
Tarjan, R.E.: Sorting using networks of queues and stacks. J. Assoc. Comput. Mach. 19, 341–346 (1972)
Wood, D.R.: Bounded degree book embeddings and three-dimensional orthogonal graph drawing. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 312–327. Springer, Heidelberg (2002)
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)
Yannakakis, M.: Embedding planar graphs in four pages. J. Comput. System Sci. 38, 36–67 (1989)
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Hasunuma, T. (2004). Laying Out Iterated Line Digraphs Using Queues. In: Liotta, G. (eds) Graph Drawing. GD 2003. Lecture Notes in Computer Science, vol 2912. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24595-7_19
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DOI: https://doi.org/10.1007/978-3-540-24595-7_19
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