Solving the 2-Disjoint Paths Problem in Nearly Linear Time

Given four distinct vertices s 1,s 2,t 1, and t 2 of a graph G, the 2-disjoint paths problem is to determine two disjoint paths, p 1 from s 1 to t 1 and p 2 from s 2 to t 2, if such paths exist. Disjoint can mean vertex- or edge-disjoint.

[1]  Alok Aggarwal,et al.  Node-Disjoint Paths on the Mesh and a New Trade-Off in VLSI Layout , 2000, SIAM J. Comput..

[2]  Paul D. Seymour.  Disjoint paths in graphs , 2006, Discret. Math..

[3]  Andreas Schwill,et al.  Nonblocking Graphs: Greedy Algorithms to Compute Disjoint Paths , 1990, STACS.

[4]  C. Pandu Rangan,et al.  Edge-disjoint paths in permutation graphs , 1994, Discuss. Math. Graph Theory.

[5]  Yehoshua Perl,et al.  Finding Two Disjoint Paths Between Two Pairs of Vertices in a Graph , 1978, JACM.

[6]  James F. Lynch,et al.  The equivalence of theorem proving and the interconnection problem , 1975, SIGD.

[7]  Robert E. Tarjan,et al.  A quick method for finding shortest pairs of disjoint paths , 1984, Networks.

[8]  Carsten Thomassen,et al.  2-Linked Graphs , 1980, Eur. J. Comb..

[9]  Roberto Tamassia,et al.  On-line maintenance of triconnected components with SPQR-trees , 1996, Algorithmica.

[10]  Samir Khuller,et al.  Processor Efficient Parallel Algorithms for the Two Disjoint Paths Problem and for Finding a Kuratowski Homeomorph , 1992, SIAM J. Comput..

[11]  Jan van Leeuwen,et al.  Worst-case Analysis of Set Union Algorithms , 1984, JACM.

[12]  Gregory Gutin,et al.  Digraphs - theory, algorithms and applications , 2002 .

[13]  Neil Robertson,et al.  Graph Minors .XIII. The Disjoint Paths Problem , 1995, J. Comb. Theory B.

[14]  C. Kuratowski.  Sur le problème des courbes gauches en Topologie , 1930 .

[15]  K. Menger.  Zur allgemeinen Kurventheorie , 1927 .

[16]  Richard M. Karp,et al.  On the Computational Complexity of Combinatorial Problems , 1975, Networks.

[17]  Roberto Tamassia,et al.  On-line maintenance of the four-connected components of a graph , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[18]  John E. Hopcroft,et al.  The Directed Subgraph Homeomorphism Problem , 1978, Theor. Comput. Sci..

[19]  Ephraim Korach,et al.  General Vertex Disjoint Paths in Series-Parallel Graphs , 1993, Discret. Appl. Math..

[20]  A. Schrijver.  A group-theoretical approach to disjoint paths in directed graphs , 1993 .

[21]  Yossi Shiloach,et al.  A Polynomial Solution to the Undirected Two Paths Problem , 1980, JACM.

[22]  Mark E. Watkins.  On the existence of certain disjoint arcs in graphs , 1968 .

[23]  S. Gill Williamson,et al.  Depth-First Search and Kuratowski Subgraphs , 1984, JACM.

[24]  H. Whitney.  Congruent Graphs and the Connectivity of Graphs , 1932 .

[25]  Bruce A. Reed,et al.  An Improved Algorithm for Finding Tree Decompositions of Small Width , 1999, Int. J. Found. Comput. Sci..

[26]  Paul D. Seymour,et al.  Graph Minors: XV. Giant Steps , 1996, J. Comb. Theory, Ser. B.

[27]  Robert E. Tarjan,et al.  Efficient Planarity Testing , 1974, JACM.

[28]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[29]  Tatsuo Ohtsuki,et al.  The two disjoint path problem and wire routing design , 1980, Graph Theory and Algorithms.

[30]  Gerhard J. Woeginger,et al.  A Simple Solution to the Two Paths Problem in Planar Graphs , 1990, Information Processing Letters.

[31]  Torsten Tholey.  Finding Disjoint Paths on Directed Acyclic Graphs , 2005, WG.

[32]  Dieter Jungnickel,et al.  Graphen, Netzwerke und Algorithmen , 1987 .

[33]  Jeffery R. Westbrook,et al.  Maintaining the Classes of 4-Edge-Connectivity in a Graph On-Line , 1998, Algorithmica.

[34]  Alon Itai,et al.  On the Complexity of Timetable and Multicommodity Flow Problems , 1976, SIAM J. Comput..