On Linear Layouts of Graphs

In a total order of the vertices of a graph, two edges with no endpoint in common can be \emphcrossing, \emphnested, or \emphdisjoint. A \emphk-stack (respectively, \emphk-queue, \emphk-arch) \emphlayout of a graph consists of a total order of the vertices, and a partition of the edges into k sets of pairwise non-crossing (non-nested, non-disjoint) edges. Motivated by numerous applications, stack layouts (also called \emphbook embeddings) and queue layouts are widely studied in the literature, while this is the first paper to investigate arch layouts.\par Our main result is a characterisation of k-arch graphs as the \emphalmost (k+1)-colourable graphs; that is, the graphs G with a set S of at most k vertices, such that G S is (k+1)-colourable.\par In addition, we survey the following fundamental questions regarding each type of layout, and in the case of queue layouts, provide simple proofs of a number of existing results. How does one partition the edges given a fixed ordering of the vertices? What is the maximum number of edges in each type of layout? What is the maximum chromatic number of a graph admitting each type of layout? What is the computational complexity of recognising the graphs that admit each type of layout?\par A comprehensive bibliography of all known references on these topics is included. \par

[1]  Hikoe Enomoto,et al.  On the Pagenumber of Complete Bipartite Graphs , 1997, J. Comb. Theory, Ser. B.

[2]  Farhad Shahrokhi,et al.  The book crossing number of a graph , 1996, J. Graph Theory.

[3]  Emilio Di Giacomo,et al.  Book Embeddings and Point-Set Embeddings of Series-Parallel Digraphs , 2002, GD.

[4]  Seth M. Malitz,et al.  Graphs with E Edges Have Pagenumber O(sqrt(E)) , 1994, J. Algorithms.

[5]  J. Ian Munro,et al.  Succinct Representation of Balanced Parentheses and Static Trees , 2002, SIAM J. Comput..

[6]  David R. Wood,et al.  Bounded Degree Book Embeddings and Three-Dimensional Orthogonal Graph Drawing , 2001, GD.

[7]  Alexandr V. Kostochka,et al.  Upper bounds on the chromatic number of graphs , 1988 .

[8]  András Gyárfás,et al.  On the chromatic number of multiple interval graphs and overlap graphs , 1985, Discret. Math..

[9]  Mihalis Yannakakis,et al.  Embedding Planar Graphs in Four Pages , 1989, J. Comput. Syst. Sci..

[10]  Lenwood S. Heath,et al.  Stack and Queue Layouts of Directed Acyclic Graphs: Part II , 1999, SIAM J. Comput..

[11]  Bogdan Oporowski,et al.  Drawing Subdivisions Of Complete And Complete Bipartite Graphs On Books , 1999 .

[12]  Toru Hasunuma,et al.  Embedding iterated line digraphs in books , 2002, Networks.

[13]  Lenwood S. Heath,et al.  The pagenumber of k-trees is O(k) , 2001, Discret. Appl. Math..

[14]  Elena Stöhr A Trade-off between Page Number and Page Width of Book Embeddings of Graphs , 1988, Inf. Comput..

[15]  Ramjee P. Swaminathan,et al.  The Pagenumber of the Class of Bandwidth-k Graphs is k-1 , 1995, Inf. Process. Lett..

[16]  Paul C. Kainen,et al.  Book embeddings of graphs and a theorem of Whitney , 2021, 2110.00820.

[17]  Arnold L. Rosenberg,et al.  Embedding graphs in books: a layout problem with applications to VLSI design , 1985 .

[18]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[19]  A. Itai,et al.  QUEUES, STACKS AND GRAPHS , 1971 .

[20]  Bojana Obrenic,et al.  Embedding de Bruijn and shuffle-exchange graphs in five pages (preliminary version) , 1991, SPAA '91.

[21]  Walter Unger,et al.  The Complexity of Colouring Circle Graphs (Extended Abstract) , 1992, STACS.

[22]  Mihalis Yannakais,et al.  Embedding planar graphs in four pages , 1989, STOC 1989.

[23]  Lenwood S. Heath,et al.  Stack and Queue Layouts of Posets , 1997, SIAM J. Discret. Math..

[24]  Patrice Ossona de Mendez,et al.  A left-first search algorithm for planar graphs , 1995, Discret. Comput. Geom..

[25]  Lenwood S. Heath,et al.  Stack and Queue Layouts of Directed Planar Graphs , 1993, Planar Graphs.

[26]  Arnold L. Rosenberg,et al.  Scheduling Tree-Dags Using FIFO Queues: A Control-Memory Trade-Off , 1996, J. Parallel Distributed Comput..

[27]  Tomasz Bilski Optimum embedding of complete graphs in books , 1998, Discret. Math..

[28]  Akihiro Nozaki,et al.  Generating and sorting permutations using restricted-deques , 1977 .

[29]  C. E. Veni Madhavan,et al.  Stack and Queue Number of 2-Trees , 1995, COCOON.

[30]  Koichi Yamazaki,et al.  Pagenumber of pathwidth-k graphs and strong pathwidth-k graphs , 2002, Discret. Math..

[31]  Peter W. Shor,et al.  On the pagenumber of planar graphs , 1984, STOC '84.

[32]  András Gyárfás,et al.  Covering and coloring problems for relatives of intervals , 1985, Discret. Math..

[33]  Arnold L. Rosenberg,et al.  Diogenes, Circa 1986 , 1986, Aegean Workshop on Computing.

[34]  Cyril Gavoille,et al.  Compact Routing Tables for Graphs of Bounded Genus , 1999, ICALP.

[35]  R. P. Dilworth,et al.  A DECOMPOSITION THEOREM FOR PARTIALLY ORDERED SETS , 1950 .

[36]  Ivan Stojmenovic,et al.  A Genetic Algorithm for Finding the Pagenumber of Interconnection Networks , 2002, J. Parallel Distributed Comput..

[37]  David R. Wood,et al.  Tree-Partitions of k-Trees with Applications in Graph Layout , 2003, WG.

[38]  Elena Stöhr The pagewidth of trivalent planar graphs , 1991, Discret. Math..

[39]  Richard A. Games Optimal book embeddings of the FFT, benes, and barrel shifter networks , 2005, Algorithmica.

[40]  R. Blankenship Book embeddings of graphs , 2003 .

[41]  David P. Dailey Uniqueness of colorability and colorability of planar 4-regular graphs are NP-complete , 1980, Discret. Math..

[42]  Alexandr V. Kostochka,et al.  The pagenumber of spherical lattices is unbounded , 2001 .

[43]  Shlomo Moran,et al.  Two-Page Book Embedding of Trees under Vertex-Neighborhood Constraints , 1993, Discret. Appl. Math..

[44]  Endre Szemerédi,et al.  On nontrivial separators for k-page graphs and simulations by nondeterministic one-tape Turing machines , 1986, STOC '86.

[45]  Joseph L. Ganley,et al.  Stack and Queue Layouts of Halin Graphs , 2001 .

[46]  Lenwood S. Heath Embedding outerplanar graphs in small books , 1987 .

[47]  Ben Dushnik,et al.  Partially Ordered Sets , 1941 .

[48]  Walter UngerFachbereich The Complexity of Colouring Circle Graphs , 1992 .

[49]  Gary L. Miller,et al.  The Complexity of Coloring Circular Arcs and Chords , 1980, SIAM J. Algebraic Discret. Methods.

[50]  Arnold L. Rosenberg,et al.  Vertex Types in Book-Embeddings , 1987, SIAM J. Discret. Math..

[51]  Endre Szemerédi,et al.  On 3-pushdown graphs with large separators , 1989, Comb..

[52]  S. Moran,et al.  One-page book embedding under vertex-neighborhood constraints , 1990, Proceedings of the 5th Jerusalem Conference on Information Technology, 1990. 'Next Decade in Information Technology'.

[53]  Lenwood S. Heath Embedding Planar Graphs in Seven Pages , 1984, FOCS.

[54]  Emilio Di Giacomo,et al.  Track Drawings of Graphs with Constant Queue Number , 2003, GD.

[55]  Paul C. Kainen,et al.  The book thickness of a graph , 1979, J. Comb. Theory, Ser. B.

[56]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[57]  Richard J. Nowakowski,et al.  Ordered sets, pagenumbers and planarity , 1989 .

[58]  Hans Jürgen Prömel,et al.  A note on triangle-free and bipartite graphs , 2002, Discret. Math..

[59]  Arnold L. Rosenberg,et al.  The Diogenes Approach to Testable Fault-Tolerant Arrays of Processors , 1983, IEEE Transactions on Computers.

[60]  Yen-Chi Chen,et al.  A Study of Typenumber in Book-Embedding , 2002, Ars Comb..

[61]  Endre Szemerédi,et al.  On Nontrivial Separators for k-Page Graphs and Simulations by Nondeterministic One-Tape Turing Machines , 1989, J. Comput. Syst. Sci..

[62]  Tero Harju,et al.  Forbidden subsequences and permutations sortable on two parallel stacks , 2001, Where Mathematics, Computer Science, Linguistics and Biology Meet.

[63]  Pat Morin,et al.  Layout of Graphs with Bounded Tree-Width , 2004, SIAM J. Comput..

[64]  Farhad Shahrokhi,et al.  On Crossing Sets, Disjoint Sets, and Pagenumber , 2000, J. Algorithms.

[65]  Lenwood S. Heath,et al.  Stack and Queue Layouts of Directed Acyclic Graphs: Part I , 1999, SIAM J. Comput..

[66]  Alexandr V. Kostochka,et al.  Covering and coloring polygon-circle graphs , 1997, Discret. Math..

[67]  David R. Wood,et al.  Degree constrained book embeddings , 2002, J. Algorithms.

[68]  Vaughan R. Pratt,et al.  Computing permutations with double-ended queues, parallel stacks and parallel queues , 1973, STOC.

[69]  Guy Jacobson,et al.  Space-efficient static trees and graphs , 1989, 30th Annual Symposium on Foundations of Computer Science.

[70]  Alexander A. Ageev,et al.  A triangle-free circle graph with chromatic number 5 , 1996, Discret. Math..

[71]  Robert E. Tarjan,et al.  Sorting Using Networks of Queues and Stacks , 1972, J. ACM.

[72]  Maciej M. Syslo,et al.  Bounds to the Page Number of Partially Ordered Sets , 1990, WG.

[73]  Arnold L. Rosenberg,et al.  Comparing Queues and Stacks as Mechanisms for Laying out Graphs , 1992, SIAM J. Discret. Math..

[74]  Seth M. Malitz,et al.  Genus g Graphs Have Pagenumber O(sqrt(g)) , 1994, J. Algorithms.

[75]  Yukio Shibata,et al.  Embedding De Bruijn, Kautz and Shuffle-exchange Networks in Books , 1997, Discret. Appl. Math..

[76]  Sriram Venkata Pemmarju Exploring the powers of stacks and queues via graph layouts , 1992 .

[77]  Toshiki Endo,et al.  The pagenumber of toroidal graphs is at most seven , 1997, Discret. Math..

[78]  Farhad Shahrokhi,et al.  The book crossing number of a graph , 1996 .

[79]  Lenwood S. Heath,et al.  Laying out Graphs Using Queues , 1992, SIAM J. Comput..

[80]  Lenwood S. Heath,et al.  The pagenumber of genus g graphs is O(g) , 1992, JACM.

[81]  David R. Wood,et al.  Stacks, Queues and Tracks: Layouts of Graph Subdivisions , 2005, Discret. Math. Theor. Comput. Sci..

[82]  S. Mitchell Linear algorithms to recognize outerplanar and maximal outerplanar graphs , 1979 .

[83]  Ravi Kannan,et al.  Unraveling k-page graphs , 1985, Inf. Control..

[84]  Robert Hochberg,et al.  Optimal one-page tree embeddings in linear time , 2003, Inf. Process. Lett..

[85]  Walter Unger,et al.  On the k-Colouring of Circle-Graphs , 1988, STACS.

[86]  David R. Wood,et al.  Queue Layouts, Tree-Width, and Three-Dimensional Graph Drawing , 2002, FSTTCS.

[87]  David Eppstein,et al.  Separating Geometric Thickness from Book Thickness , 2001, ArXiv.

[88]  Ivan Rival,et al.  Series-Parallel Planar Ordered Sets Have Pagenumber Two , 1996, GD.

[89]  増山 繁,et al.  Deciding whether Graph $G$ Has Page Number One is in NC , 1992 .

[90]  Prosenjit Bose,et al.  The Maximum Number of Edges in a Three-Dimensional Grid-Drawing , 2004, J. Graph Algorithms Appl..

[91]  David R. Wood,et al.  Track Layouts of Graphs , 2004, Discret. Math. Theor. Comput. Sci..

[92]  Douglas B. West,et al.  Pagenumber of complete bipartite graphs , 1988, J. Graph Theory.

[93]  李幼升,et al.  Ph , 1989 .

[94]  O. D'Antona,et al.  Book-thickness and book-coarseness of graphs , 1984 .

[95]  Elena Stöhr Optimal Book Embeddings of Depth-k Kn-cylinders , 1990, J. Inf. Process. Cybern..

[96]  Hung-Lin Fu,et al.  The typenumber of trees , 2002, Discret. Math..

[97]  Emilio Di Giacomo,et al.  Drawing Planar Graphs on a Curve , 2003, WG.

[98]  S. M. Malitz Graphs with E edges have pagenumber O(√E) , 1994, FOCS 1994.