Interval Graphs: Canonical Representations in Logspace
Abstract
We present a logspace algorithm for computing a canonical labeling, in fact, a canonical interval representation, for interval graphs. To achieve this, we compute canonical interval representations of interval hypergraphs. This approach also yields a canonical labeling of convex graphs. As a consequence, the isomorphism and automorphism problems for these graph classes are solvable in logspace. For proper interval graphs we also design logspace algorithms computing their canonical representations by proper and by unit interval systems.
MSC codes
Keywords
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
F. Annexstein and R. Swaminathan, On testing consecutive-ones property in parallel, Discrete Appl. Math., 88 (1998), pp. 7–28.
2.
L. Babel, I. N. Ponomarenko, and G. Tinhofer, The isomorphism problem for directed path graphs and for rooted directed path graphs, J. Algorithms, 21 (1996), pp. 542–564.
3.
J. Bang-Jensen, J. Huang, and L. Ibarra, Recognizing and representing proper interval graphs in parallel using merging and sorting, Discrete Appl. Math., 155 (2007), pp. 442–456.
4.
K. S. Booth and G. S. Lueker, Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms, J. Comput. System Sci., 13 (1976), pp. 335–379.
5.
A. Brandstädt, V. B. Le, and J. P. Spinrad, Graph Classes: A Survey, SIAM Monogr. Discrete Math. Appl. 3, SIAM, Philadelphia, 1999.
6.
J. Cai, M. Fürer, and N. Immerman An optimal lower bound on the number of variables for graph identification, Combinatorica, 12 (1992), pp. 389–410.
7.
L. Chen, Graph isomorphism and identification matrices: Parallel algorithms, IEEE Trans. Parallel Distrib. Syst., 7 (1996), pp. 308–319.
8.
L. Chen, Graph isomorphism and identification matrices: Sequential algorithms, J. Comput. System Sci., 59 (1999), pp. 450–475.
9.
D. G. Corneil, H. Kim, S. Natarajan, S. Olariu, and A. P. Sprague, Simple linear time recognition of unit interval graphs, Inform. Process. Lett., 55 (1995), pp. 99–104.
10.
L. Chen and Y. Yesha, Parallel recognition of the consecutive-ones property with applications, J. Algorithms, 12 (1991), pp. 375–392.
11.
L. Chen and Y. Yesha, Efficient parallel algorithms for bipartite permutation graphs, Networks, 23 (1993), pp. 29–39.
12.
X. Deng, P. Hell, and J. Huang, Linear-time representation algorithms for proper circular-arc graphs and proper interval graphs, SIAM J. Comput., 25 (1996), pp. 390–403.
13.
S. Datta, P. Nimbhorkar, T. Thierauf, and F. Wagner, Graph isomorphism for $K_{3,3}$-free and $K_5$-free graphs is in log-space, in FSTTCS, Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2009, pp. 145–156.
14.
S. Datta, N. Limaye, P. Nimbhorkar, T. Thierauf, and F. Wagner, Planar graph isomorphism is in log-space, in CCC, IEEE Computer Society, Washington, DC, 2009, pp. 203–214.
15.
P. Duchet, Propriété de helly et problèmes de représentation, in Problèmes Combinatoires et Théorie des Graphes (Orsay, 1976), Colloq. Int. CNRS 260, CNRS, Paris, France, 1978, pp. 117–118.
16.
P. Duchet, Classical perfect graphs, Ann. Discrete Math., 21 (1984), pp. 67–96.
17.
K. Etessami, Counting quantifiers, successor relations, and logarithmic space, J. Comput. System Sci., 54 (1997), pp. 400–411.
18.
S. Evdokimov, I. N. Ponomarenko, and G. Tinhofer Forestal algebras and algebraic forests (on a new class of weakly compact graphs), Discrete Math., 225 (2000), pp. 149–172.
19.
D. R. Fulkerson and O. A. Gross, Incidence matrices and interval graphs, Pacific J. Math., 15 (1965), pp. 835–855.
20.
M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, 2nd ed., Ann. Discrete Math. 57, Elsevier, New York, 2004.
21.
M. Grohe and O. Verbitsky, Testing graph isomorphism in parallel by playing a game, in ICALP, Springer, Berlin, 2006, pp. 3–14.
22.
M. Habib, R. M. McConnell, C. Paul, and L. Viennot, Lex-BFS and partition refinement, Theoret. Comput. Sci., 234 (2000), pp. 59–84.
23.
F. Harary and T. A. McKee, The square of a chordal graph, Discrete Math., 128 (1994), pp. 165–172.
24.
W.-L. Hsu and R. M. McConnell, PC trees and circular-ones arrangements, Theoret. Comput. Sci., 296 (2003), pp. 99–116
25.
P. Hell, R. Shamir, and R. Sharan, A fully dynamic algorithm for recognizing and representing proper interval graphs, SIAM J. Comput., 31 (2001), pp. 289–305.
26.
C. M. Hoffmann, Group-Theoretic Algorithms and Graph Isomorphism, Lecture Notes in Comput. Sci. 136, Springer, Berlin, 1982.
27.
W.-L. Hsu, $O(M\cdotN)$ algorithms for the recognition and isomorphism problems on circular-arc graphs, SIAM J. Comput., 24 (1995), pp. 411–439.
28.
W.-L. Hsu and T.-H. Ma, Fast and simple algorithms for recognizing chordal comparability graphs and interval graphs, SIAM J. Comput., 28 (1999), pp. 1004–1020.
29.
P. N. Klein, Efficient parallel algorithms for chordal graphs, SIAM J. Comput., 25 (1996), pp. 797–827.
30.
J. Köbler and S. Kuhnert, The isomorphism problem for k-trees is complete for logspace, in MFCS, Springer, Berlin, 2009, pp. 537–548.
31.
P. N. Klein and J. H. Reif, An efficient parallel algorithm for planarity, J. Comput. System Sci., 37 (1988), pp. 190–246.
32.
J. Köbler, U. Schöning, and J. Torán, The Graph Isomorphism Problem: Its Structural Complexity, Progr. Theoret. Comput. Sci., Birkhäuser Boston, Boston, 1993.
33.
D. Kozen, U. V. Vazirani, and V. V. Vazirani, NC algorithms for comparability graphs, interval graphs, and testing for unique perfect matching, in FSTTCS, Lecture Notes in Comput. Sci. 206, Springer, Berlin, 1985, pp. 496–503.
34.
J. Kratochvíl, A special planar satisfiability problem and a consequence of its NP–completeness, Discrete Appl. Math., 52 (1994), pp. 233–252.
35.
J. Kratochvíl, Intersection graphs of noncrossing arc–connected sets in the plane, in GD, Springer, Berlin, 1996, pp. 257–270.
36.
B. Laubner, Capturing polynomial time on interval graphs, in LICS, IEEE Computer Society, Washington, DC, 2010, pp. 199–208.
37.
S. Lindell, A logspace algorithm for tree canonization, in STOC, ACM, New York, 1992, pp. 400–404.
38.
G. S. Lueker and K. S. Booth, A linear time algorithm for deciding interval graph isomorphism, J. ACM, 26 (1979), pp. 183–195.
39.
R. M. McConnell, Linear-time recognition of circular-arc graphs, Algorithmica, 37 (2003), pp. 93–147.
40.
J. H. Reif, Symmetric complementation, J. ACM, 31 (1984), pp. 401–421.
41.
O. Reingold, Undirected connectivity in log-space, J. ACM, 55 (2008), pp. 17.1–17.24.
42.
F. S. Roberts, Indifference graphs, in Proof Techniques in Graph Theory: Proceedings of the 2nd Annual Arbor Graph Theory Conference, Academic Press, New York, 1969, pp. 139–146.
43.
D. J. Rose, R. E. Tarjan, and G. S. Lueker, Algorithmic aspects of vertex elimination on graphs, SIAM J. Comput., 5 (1976), pp. 266–283.
44.
J. P. Spinrad, Efficient Graph Representations, Fields Inst. Monogr. 19, AMS, Providence, RI, 2003.
45.
A. Tucker, A structure theorem for the consecutive 1's property, J. Combin. Theory Ser. B, 12 (1972), pp. 153–162.
46.
R. Uehara, Simple geometrical intersection graphs, in WALCOM, Lecture Notes in Comput. Sci. 4921, Springer, Berlin, 2008, pp. 25–33.
47.
C.-W. Yu and G.-H. Chen, An efficient parallel recognition algorithm for bipartite-permutation graphs, IEEE Trans. Parallel Distrib. Syst., 7 (1996), pp. 3–10.
Information & Authors
Information
Published In
Copyright
Copyright © 2011 Society for Industrial and Applied Mathematics.
History
Submitted: 29 July 2010
Accepted: 29 June 2011
Published online: 20 September 2011
MSC codes
Keywords
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Compact representation of interval graphs and circular-arc graphs of bounded degree and chromatic numberTheoretical Computer Science, Vol. 941 | 1 Jan 2023
- Planar Graph Isomorphism is in Log-SpaceACM Transactions on Computation Theory, Vol. 204 | 26 July 2022
- Cyclic arrangements with minimum modulo m winding numbersGraphs and Combinatorics, Vol. 38, No. 3 | 14 March 2022
- Compact Representation of Interval Graphs of Bounded Degree and Chromatic Number2022 Data Compression Conference (DCC) | 1 Mar 2022
- The Weisfeiler–Leman Dimension of Chordal Bipartite Graphs Without Bipartite ClawGraphs and Combinatorics, Vol. 37, No. 3 | 25 March 2021
- Graph isomorphism restricted by listsTheoretical Computer Science, Vol. 860 | 1 Mar 2021
- Canonical Representations for Circular-Arc Graphs Using Flip SetsAlgorithmica, Vol. 80, No. 12 | 23 January 2018
- On the Parallel Parameterized Complexity of the Graph Isomorphism ProblemWALCOM: Algorithms and Computation | 31 January 2018
- On Compiling Structured CNFs to OBDDsTheory of Computing Systems, Vol. 61, No. 2 | 26 October 2016
- Circular-arc hypergraphs: Rigidity via connectednessDiscrete Applied Mathematics, Vol. 217 | 1 Jan 2017
- Solving the canonical representation and Star System Problems for proper circular-arc graphs in logspaceJournal of Discrete Algorithms, Vol. 38-41 | 1 May 2016
- On the isomorphism problem for Helly circular-arc graphsInformation and Computation, Vol. 247 | 1 Apr 2016
- Interval graph representation with given interval and intersection lengthsJournal of Discrete Algorithms, Vol. 34 | 1 Sep 2015
- On Compiling Structured CNFs to OBDDsComputer Science -- Theory and Applications | 23 June 2015
- Helly Circular-Arc Graph Isomorphism Is in LogspaceMathematical Foundations of Computer Science 2013 | 1 Jan 2013
- Interval Graph Representation with Given Interval and Intersection LengthsAlgorithms and Computation | 1 Jan 2012