Around and Beyond the Isomorphism Problem for Interval Graphs

The class of problems solvable in logarithmic space has recently replenished with the isomorphism testing for interval graphs. We discuss this result, prospects of its extension to larger classes of graphs, and related issues such as constructing canonical models of intersection graphs and solving the Star System Problem for restricted classes of graphs.

[1]  Jayme Luiz Szwarcfiter,et al.  Linear-Time Recognition of Helly Circular-Arc Models and Graphs , 2011, Algorithmica.

[2]  Eberhard Triesch,et al.  Reconstructing a Graph from its Neighborhood Lists , 1993, Comb. Probab. Comput..

[3]  Dániel Marx,et al.  Structure theorem and isomorphism test for graphs with excluded topological subgraphs , 2011, STOC '12.

[4]  Jacobo Torán,et al.  Completeness results for graph isomorphism , 2003, J. Comput. Syst. Sci..

[5]  Philip N. Klein.  Efficient Parallel Algorithms for Chordal Graphs , 1996, SIAM J. Comput..

[6]  Lin Chen.  Graph Isomorphism and Identification Matrices: Sequential Algorithms , 1999, J. Comput. Syst. Sci..

[7]  J. Moon,et al.  On cliques in graphs , 1965 .

[8]  Georg Gati,et al.  Further annotated bibliography on the isomorphism disease , 1979, J. Graph Theory.

[9]  Stephan Olariu,et al.  Simple Linear Time Recognition of Unit Interval Graphs , 1995, Inf. Process. Lett..

[10]  Jacobo Torán,et al.  Restricted space algorithms for isomorphism on bounded treewidth graphs , 2010, Inf. Comput..

[11]  Xiaotie Deng,et al.  Linear-Time Representation Algorithms for Proper Circular-Arc Graphs and Proper Interval Graphs , 1996, SIAM J. Comput..

[12]  Andreas Blass,et al.  On the Unique Satisfiability Problem , 1982, Inf. Control..

[13]  László Babai,et al.  Canonical labeling of graphs , 1983, STOC.

[14]  Oleg Verbitsky,et al.  Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Logspace , 2012, FSTTCS.

[15]  P. Duchet.  Classical Perfect Graphs: An introduction with emphasis on triangulated and interval graphs , 1984 .

[16]  D. Corneil,et al.  Isomorphism Testing in Hookup Classes , 1982 .

[17]  Seinosuke Toda.  Computing Automorphism Groups of Chordal Graphs Whose Simplicial Components Are of Small Size , 2006, IEICE Trans. Inf. Syst..

[18]  Ryuhei Uehara,et al.  Simple Geometrical Intersection Graphs , 2008, WALCOM.

[19]  Jayme Luiz Szwarcfiter,et al.  Characterizations and recognition of circular-arc graphs and subclasses: A survey , 2009, Discret. Math..

[20]  Kellogg S. Booth,et al.  Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms , 1976, J. Comput. Syst. Sci..

[21]  Johannes Köbler,et al.  On Graph Isomorphism for Restricted Graph Classes , 2006, CiE.

[22]  Gottfried Tinhofer,et al.  The Isomorphism Problem For Directed Path Graphs and For Rooted Directed Path Graphs , 1996, J. Algorithms.

[23]  Jayme Luiz Szwarcfiter,et al.  Isomorphism of graph classes related to the circular-ones property , 2012, Discret. Math. Theor. Comput. Sci..

[24]  Lin Chen.  Graph Isomorphism and Identification Matrices: Parallel Algorithms , 1996, IEEE Trans. Parallel Distributed Syst..

[25]  I. S. Filotti,et al.  A Polynomial-time Algorithm for Determining the Isomorphism of Graphs of Fixed Genus (Working Paper) , 1980, STOC 1980.

[26]  Fanica Gavril,et al.  Algorithms on circular-arc graphs , 1974, Networks.

[27]  Eugene M. Luks,et al.  Isomorphism of graphs of bounded valence can be tested in polynomial time , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[28]  Hans L. Boblaender.  Polynomial algorithms for graph isomorphism and chromatic index on partial k -trees , 1990 .

[29]  Koichi Yamazaki,et al.  Isomorphism for Graphs of Bounded Distance Width , 1997, Algorithmica.

[30]  Jørgen Bang-Jensen,et al.  Convex-Round and Concave-Round Graphs , 2000, SIAM J. Discret. Math..

[31]  Terry A. McKee,et al.  The square of a chordal graph , 1994, Discret. Math..

[32]  László Babai,et al.  Isomorhism of Hypergraphs of Low Rank in Moderately Exponential Time , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[33]  Bastian Laubner,et al.  Capturing Polynomial Time on Interval Graphs , 2009, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[34]  Steven Lindell.  A Logspace Algorithm for Tree Canonization , 1992 .

[35]  John E. Hopcroft,et al.  Linear time algorithm for isomorphism of planar graphs (Preliminary Report) , 1974, STOC '74.

[36]  Oleg Verbitsky,et al.  Testing Graph Isomorphism in Parallel by Playing a Game , 2006, ICALP.

[37]  Vikraman Arvind,et al.  The isomorphism problem for k-trees is complete for logspace , 2012, Inf. Comput..

[38]  Benedikt Löwe,et al.  Logical Approaches to Computational Barriers: CiE 2006 , 2007, J. Log. Comput..

[39]  Robert E. Tarjan,et al.  A V² Algorithm for Determining Isomorphism of Planar Graphs , 1971, Inf. Process. Lett..

[40]  Martin Grohe,et al.  Isomorphism testing for embeddable graphs through definability , 2000, STOC '00.

[41]  A. Tucker,et al.  Matrix characterizations of circular-arc graphs , 1971 .

[42]  Jacobo Torán,et al.  The Complexity of Planar Graph Isomorphism , 2009, Bull. EATCS.

[43]  Stephan Olariu,et al.  On the Isomorphism of Graphs with Few P4s , 1995, WG.

[44]  Derek G. Corneil,et al.  The graph isomorphism disease , 1977, J. Graph Theory.

[45]  Gary L. Miller,et al.  Isomorphism testing for graphs of bounded genus , 1980, STOC '80.

[46]  Wen-Lian Hsu,et al.  PC trees and circular-ones arrangements , 2003, Theor. Comput. Sci..

[47]  Jacobo Torán,et al.  Isomorphism Testing: Perspective and Open Problems , 2005, Bull. EATCS.

[48]  Jacobo Torán.  On the Hardness of Graph Isomorphism , 2004, SIAM J. Comput..

[49]  Oleg Verbitsky,et al.  Interval Graphs: Canonical Representations in Logspace , 2010, SIAM J. Comput..

[50]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

[51]  Vladimir Gurvich,et al.  Neighborhood hypergraphs of bipartite graphs , 2008 .

[52]  Pascal Schweitzer.  Isomorphism of (mis)Labeled Graphs , 2011, ESA.

[53]  Kellogg S. Booth,et al.  A Linear Time Algorithm for Deciding Interval Graph Isomorphism , 1979, JACM.

[54]  Stefan Kratsch,et al.  Isomorphism for Graphs of Bounded Feedback Vertex Set Number , 2010, SWAT.

[55]  Ken-ichi Kawarabayashi,et al.  Graph and map isomorphism and all polyhedral embeddings in linear time , 2008, STOC.

[56]  Lin Chen,et al.  Parallel Recognition of the Consecutive Ones Property with Applications , 1991, J. Algorithms.

[57]  Omer Reingold,et al.  Undirected connectivity in log-space , 2008, JACM.

[58]  Fedor V. Fomin,et al.  On the complexity of reconstructing H-free graphs from their Star Systems , 2011, J. Graph Theory.

[59]  Sergei Evdokimov,et al.  Isomorphism of Coloured Graphs with Slowly Increasing Multiplicity of Jordan Blocks , 1995, Comb..

[60]  Regina Tyshkevich,et al.  Graph isomorphism problem , 1985 .

[61]  Gary L. Miller,et al.  Parallel Tree Contraction, Part 2: Further Applications , 1991, SIAM J. Comput..

[62]  S. Benzer.  ON THE TOPOLOGY OF THE GENETIC FINE STRUCTURE. , 1959, Proceedings of the National Academy of Sciences of the United States of America.

[63]  Michael Dom,et al.  Algorithimic Aspects of the Consecutive-Ones Property , 2009, Bull. EATCS.

[64]  Samir Datta,et al.  Graph Isomorphism for K_{3, 3}-free and K_5-free graphs is in Log-space , 2009, FSTTCS.

[65]  Eugene M. Luks,et al.  Hypergraph isomorphism and structural equivalence of Boolean functions , 1999, STOC '99.

[66]  Ryuhei Uehara,et al.  Graph isomorphism completeness for chordal bipartite graphs and strongly chordal graphs , 2005, Discret. Appl. Math..

[67]  J. Köbler,et al.  The Graph Isomorphism Problem: Its Structural Complexity , 1993 .

[68]  Vikraman Arvind,et al.  Colored Hypergraph Isomorphism is Fixed Parameter Tractable , 2010, FSTTCS.

[69]  François Lalonde.  Le probleme d'etoiles pour graphes est np-complet , 1981, Discret. Math..

[70]  Nutan Limaye,et al.  Planar Graph Isomorphism is in Log-Space , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[71]  Lin Chen.  A selected tour of the theory of identification matrices , 2000, Theor. Comput. Sci..