# Encyclopedia of Algorithms

Living Edition
| Editors: Ming-Yang Kao

# Split Decomposition via Graph-Labelled Trees

Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27848-8_686-1

## Problem Definition

Years and Authors of Summarized Original Work

2012; Gioan, Paul

2014; Gioan, Paul, Tedder, Corneil

Introduced by Cunningham and Edmonds [ 11], the split decomposition, also known as the join (or 1-join) decomposition, ranges among the classical graph decomposition schemes. Given a graph G = ( V, E), a bipartition ( A, B) of the vertex set V (with $$\vert A\vert \geqslant 2$$

## Keywords

Split decomposition LexBFS Circle graphs Distance hereditary graphs Permutation graphs Parity graphs
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## References

1. 1.
Bandelt H-J, Mulder HM (1986) Distance hereditary graphs. J Comb Theory Ser B 41:182–208
2. 2.
Bouchet A (1987) Reducing prime graphs and recognizing circle graphs. Combinatorica 7:243–254
3. 3.
Bretscher A, Corneil D, Habib M, Paul C (2008) A simple linear time lexbfs cograph recognition algorithm. SIAM J Discret Math 22(4):1277–1296
4. 4.
Burlet M, Uhry JP (1984) Parity graphs. Ann Discret Math 21:253–277
5. 5.
Charbit P, de Montgolfier F, Raffinot M (2012) Linear time split decomposition revisited. SIAM J Discret Math 26(2):499–514
6. 6.
Chudnovsky M, Robertson N, Seymour P, Thomas R (2006) The strong perfect graph theorem. Ann Math 161:51–229
7. 7.
Cicerone S, Di Stefano G (1999) On the extension of bipartite to parity graphs. Discret Appl Math 95:181–195
8. 8.
Corneil D, Lerchs H, Stewart-Burlingham LK (1981) Complement reducible graphs. Discret Appl Math 3(1):163–174
9. 9.
Corneil D, Habib M, Lanlignel JM, Reed B, Rotics U (2012) Polynomial-time recognition of clique-width 3 graphs. Discret Appl Math 160(6):834–865
10. 10.
Courcelle B, Engelfriet J, Rozenberg G (1993) Handle rewriting hypergraph grammars. J Comput Syst Sci 46:218–270
11. 11.
Cunningham WH, Edmonds J (1980) A combinatorial decomposition theory. Can J Math 32(3):734–765
12. 12.
Dahlhaus E (1994) Efficient parallel and linear time sequential split decomposition (extended abstract). In: Foundations of software technology and theoretical computer science – FSTTCS, Madras. Volume 880 of lecture notes in computer science, pp 171–180Google Scholar
13. 13.
Damiand G, Habib M, Paul C (2001) A simple paradigm for graph recognition: application to cographs and distance hereditary graphs. Theor Comput Sci 263:99–111
14. 14.
Gabor CP, Hsu WL, Suppovit KJ (1989) Recognizing circle graphs in polynomial time. J ACM 36:435–473
15. 15.
Gabow H, Tarjan R (1983) A linear-time algorithm for a special case of disjoint set union. In: Annual ACM symposium on theory of computing (STOC), Boston, pp 246–251Google Scholar
16. 16.
Gioan E, Paul C (2012) Split decomposition and graph-labelled trees: characterizations and fully dynamic algorithms for totally decomposable graphs. Discret Appl Math 160(6):708–733
17. 17.
Gioan E, Paul C, Tedder M, Corneil D (2013) Circle graph recognition in time $$O(n + m)\alpha (n + m)$$. Algorithmica 69(4): 759–788 (2014)
18. 18.
Gioan E, Paul C, Tedder M, Corneil D (2013) Practical split-decomposition via graph-labelled trees. Algorithmica 69(4): 789–843 (2014)
19. 19.
Habib M, Paul C (2010) A survey on algorithmic aspects of modular decomposition. Comput Sci Rev 4:41–59
20. 20.
Habib M, McConnell RM, Paul C, Viennot L (2000) Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theor Comput Sci 234:59–84
21. 21.
Hammer P, Maffray F (1990) Completely separable graphs. Discret Appl Math 27:85–99
22. 22.
Korte N, Möhring R (1989) An incremental linear-time algorithm for recongizing interval graphs. SIAM J Comput 18(1):68–81
23. 23.
Ma T-H, Spinrad J (1994) An O(n 2) algorithm for undirected split decomposition. J Algorithms 16:145–160
24. 24.
Oum SI (2005) Graphs of bounded rank-width. PhD thesis, Princeton UniversityGoogle Scholar
25. 25.
Rose DJ, Tarjan RE, Lueker GS (1976) Algorithmic aspects of vertex elimination on graphs. SIAM J Comput 5(2):266–283
26. 26.
Spinrad J (1989) Prime testing for the split decomposition of a graph. SIAM J Discret Math 2(4):590–599
27. 27.
Spinrad J (1994) Recognition of circle graphs. J Algorithms 16:264–282
28. 28.
Trotignon N (2013) Perfect graphs: a survey. Technical report 1301.5149, arxivGoogle Scholar