Crossing-Optimal Acyclic HP-Completion for Outerplanar st-Digraphs

Given an embedded planar acyclic digraph G , the acyclic hamiltonian path completion with crossing minimization (Acyclic- HPCCM) problem is to determine a hamiltonian path completion set of edges such that, when these edges are embedded on G , they create the smallest possible number of edge crossings and turn G to a hamiltonian acyclic digraph. In this paper, we present a linear time algorithm which solves the Acyclic-HPCCM problem on any outerplanar st -digraph G . The algorithm is based on properties of the optimal solution and an st-polygon decomposition of G . As a consequence of our result, we can obtain for the class of outerplanar st -digraphs upward topological 2-page book embeddings with minimum number of spine crossings.

[1]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[2]  Charles J. Colbourn,et al.  Minimizing setups in ordered sets of fixed width , 1985 .

[3]  Hikoe Enomoto,et al.  Lower Bounds for the Number of Edge-crossings Over the Spine in a Topological Book Embedding of a Graph , 1999, Discret. Appl. Math..

[4]  Shin-Ichi Nakano,et al.  Constant Time Generation of Linear Extensions , 2005, WALCOM.

[5]  Roberto Tamassia,et al.  Algorithms for Plane Representations of Acyclic Digraphs , 1988, Theor. Comput. Sci..

[6]  Emilio Di Giacomo,et al.  Curve-constrained drawings of planar graphs , 2005, Comput. Geom..

[7]  Jutta Mitas.  Tackling the jump number of interval orders , 1991 .

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

[9]  Nejib Zaguia,et al.  On minimizing jumps for ordered sets , 1990 .

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

[11]  Antonios Symvonis,et al.  Optimal Acyclic Hamiltonian Path Completion for Outerplanar Triangulated st-Digraphs (with Application to Upward Topological Book Embeddings) , 2008, ArXiv.

[12]  Roberto Tamassia,et al.  Dynamic maintenance of planar digraphs, with applications , 1990, Algorithmica.

[13]  Michel Habib,et al.  Nombre de sauts et graphes série-parallèles , 1979, RAIRO Theor. Informatics Appl..

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

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

[16]  Hesham Hassan Ali,et al.  Minimizing setups in precedence constrained scheduling , 1988 .

[17]  Roberto Tamassia,et al.  A unified approach to visibility representations of planar graphs , 1986, Discret. Comput. Geom..

[18]  Antonios Symvonis,et al.  Computing upward topological book embeddings of upward planar digraphs , 2007 .

[19]  P. Winkler,et al.  Minimizing setups for cycle-free ordered sets , 1982 .

[20]  Antonios Symvonis,et al.  Computing upward topological book embeddings of upward planar digraphs , 2007, J. Discrete Algorithms.

[21]  Antonios Symvonis,et al.  Crossing-Free Acyclic Hamiltonian Path Completion for Planar st-Digraphs , 2009, ISAAC.

[22]  Frank Ruskey,et al.  Generating Linear Extensions Fast , 1994, SIAM J. Comput..

[23]  George Steiner,et al.  A linear time algorithm to find the jump number of 2-dimensional bipartite partial orders , 1987 .

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

[25]  Antonios Symvonis,et al.  Crossing-Optimal Acyclic HP-Completion for Outerplanar st-Digraphs , 2011, J. Graph Algorithms Appl..

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

[27]  Frank Harary,et al.  Graph Theory , 2016 .

[28]  David Kelly.  Fundamentals of planar ordered sets , 1987, Discret. Math..

[29]  David S. Johnson,et al.  Some simplified NP-complete problems , 1974, STOC '74.

[30]  W. Poguntke,et al.  OPTIMAL LINEAR EXTENSIONS BY INTERCHANGING CHAINS , 2010 .

[31]  Michel Habib,et al.  Jump number of dags having Dilworth number 2 , 1984, Discret. Appl. Math..

[32]  G. Brightwell,et al.  Counting linear extensions , 1991 .

[33]  Antonios Symvonis,et al.  Crossing-Optimal Acyclic Hamiltonian Path Completion and Its Application to Upward Topological Book Embeddings , 2009, WALCOM.

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

[35]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .