Book Embeddings and Point-Set Embeddings of Series-Parallel Digraphs

An optimal O(n)-time algorithm to compute an upward two-page book embedding of a series-parallel digraph with n vertices is presented. A previous algorithm of Alzohairi and Rival [1] runs in O(n3) time and assumes that the input series-parallel digraph does not have transitive edges. One consequence of our result is that series-parallel (undirected) graphs are necessarily sub-hamiltonian. This extends a previous result by Chung, Leighton, and Rosenberg [5] who proved sub-hamiltonicity for a subset of planar series-parallel graphs. Also, this paper investigates the problem of mapping planar digraphs onto a given set of points in the plane, so that the edges are drawn upward planar. This problem is called the upward point-set embedding problem. The equivalence between the problem of computing an upward two-page book embedding and an upward point-set embedding with at most one bend per edge on any given set of points is proved. An O(n log n)-time algorithm for computing an upward point-set embedding with at most one bend per edge on any given set of points for planar series-parallel digraphs is presented.

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

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

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

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

[5]  Eugene L. Lawler,et al.  The recognition of Series Parallel digraphs , 1979, SIAM J. Comput..

[6]  R. Pollack,et al.  Embedding a planar triangulation with vertices at specified points , 1991 .

[7]  Prosenjit Bose.  On Embedding an Outer-Planar Graph in a Point Set , 1997, Graph Drawing.

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

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

[10]  Ioannis G. Tollis,et al.  Graph Drawing , 1994, Lecture Notes in Computer Science.

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

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

[13]  Walter Didimo,et al.  Orthogonal and Quasi-upward Drawings with Vertices of Prescribed Size , 1999, GD.

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

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

[16]  Robert E. Tarjan,et al.  Computing an st -Numbering , 1976, Theor. Comput. Sci..

[17]  Hazel Everett,et al.  A Visibility Representation for Graphs in Three Dimensions , 1998, J. Graph Algorithms Appl..

[18]  Roberto Tamassia,et al.  On-Line Planarity Testing , 1989, SIAM J. Comput..