Split graph

In graph theory, a split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by Földes and Hammer in two papers in 1977, and independently introduced by Tyshkevich and Chernyak (1979). [Földes and Hammer (1977a) had a more general definition, in which the graphs they called split graphs also included bipartite graphs (that is, graphs that be partitioned into two independent sets) and the complements of bipartite graphs (that is, graphs that can be partitioned into two cliques). Földes and Hammer (1977b) use the definition given here, which has been followed in the subsequent literature; for instance, it is Brandstädt et al. (1999), Definition 3.2.3, p.41. ]

Note that the partition into a clique and an independent set need not be unique; for instance, the path "a"–"b"–"c" is a split graph, the vertices of which can be partitioned in three different ways:
#the clique {"a","b"} and the independent set {"c"}
#the clique {"b","c"} and the independent set {"a"}
#the clique {"b"} and the independent set {"a","c"}

Split graphs can be characterized in terms of their induced subgraphs: a graph is split if and only if no induced subgraph is a cycle on four or five vertices, or a pair of disjoint edges (the complement of a 4-cycle). [Földes and Hammer (1977a); Golumbic (1980), Theorem 6.3, p. 151.]

Relation to other graph families

From the definition, split graphs are clearly closed under complementation. [Golumbic (1980), Theorem 6.1, p. 150.] Another characterization of split graphs involves complementation: they are chordal graphs the complements of which are also chordal. [Földes and Hammer (1977a); Golumbic (1980), Theorem 6.3, p. 151; Brandstädt et al. (1999), Theorem 3.2.3, p. 41.] Just as chordal graphs are the intersection graphs of subtrees of trees, split graphs are the intersection graphs of distinct substars of stars. [McMorris and Shier (1983); Voss (1985); Brandstädt et al. (1999), Theorem 4.4.2, p. 52.] Almost all chordal graphs are split graphs, as Bender et al. (1985) showed; that is, in the limit as "n" goes to infinity, the fraction of "n"-vertex chordal graphs that are split approaches one. Because chordal graphs are perfect, so are the split graphs; the double split graphs, a family of graphs derived from split graphs by doubling every vertex (so the clique comes to induce an antimatching and the independent set comes to induce a matching), figure prominently as one of five basic classes of perfect graphs from which all others can be formed in the proof by Chudnovsky et al. (2006) of the strong perfect graph theorem.

If a graph is both a split graph and an interval graph, then its complement is both a split graph and a comparability graph, and vice versa. The split comparability graphs, and therefore also the split interval graphs, can be characterized in terms of a set of three forbidden induced subgraphs. [Földes and Hammer (1977b); Golumbic (1980), Theorem 9.7, page 212.] The split cographs are exactly the threshold graphs, and the split permutation graphs are exactly the interval graphs that have interval graph complements. [Brandstädt et al. (1999), Corollary 7.1.1, p. 106, and Theorem 7.1.2, p. 107.] Split graphs have cochromatic number 2.

Maximum clique and maximum independent set

Let "G" be a split graph, partitioned into a clique "C" and an independent set "I". Then every maximal clique in a split graph is either "C" itself, or the neighborhood of a vertex in "I". Thus, it is easy to identify the maximum clique, and complementarily the maximum independent set in a split graph. In any split graph, one of the following three possibilities must be true: [Hammer and Simeone (1981); Golumbic (1980), Theorem 6.2, p. 150.]
# There exists a vertex "x" in "I" such that "C" ∪ {"x"} is complete. In this case, "C" ∪ {"x"} is a maximum clique and "I" is a maximum independent set.
# There exists a vertex "x" in "C" such that "I" ∪ {"x"} is independent. In this case, "I" ∪ {"x"} is a maximum independent set and "C" is a maximum clique.
# "C" is a maximal clique and "I" is a maximal independent set. In this case, "G" has a unique partition ("C","I") into a clique and an independent set, "C" is the maximum clique, and "I" is the maximum independent set.

Some other optimization problems that are NP-complete on more general graph families, including graph coloring, are similarly straightforward on split graphs.

Degree sequences

One remarkable property of split graphs is that they can be recognized solely from the sequence of degrees of each vertex. That is, suppose that two graphs have the property that, for each "i", both graphs have equally many vertices with "i" neighboring edges; then either both graphs are split or both are not split. More precisely, sort the degrees of the vertices in a graph "G" into the ordered sequence "d"₁ ≥ "d"₂ ≥ ... ≥ "d"_"n", and let "m" be the largest value of "i" such that "d"_"i" ≥ "i" - 1. Then "G" is a split graph if and only if:$sum\_\{i=1\}^m\; d\_i\; =\; m(m-1)\; +\; sum\_\{i=m+1\}^n\; d\_i.$If this is the case, then the "m" vertices with the largest degrees form a maximum clique in "G", and the remaining vertices complete a partition into a clique and an independent set. [Hammer and Simeone (1981); Tyshkevich (1980); Tyshkevich, Melnikow, and Kotov (1981); Golumbic (1980), Theorem 6.7 and Corollary 6.8, p. 154; Brandstädt et al. (1999), Theorem 13.3.2, p. 203. Merris (2003) further investigates the degree sequences of split graphs.]

Counting split graphs

Royle (2000) showed that split graphs with "n" unlabeled vertices are in one-to-one correspondence with certain Sperner families. Using this correspondence, he was able to determine a formula for the number of nonisomorphic split graphs on "n" vertices. For small values of "n", starting from "n" = 1, these numbers are:1, 2, 4, 9, 21, 56, 164, 557, 2223, 10766, 64956, 501696, ... OEIS|id = A048194. This result was independently proved earlier by Tyshkevich and Chernyak in 1990 (in Russian).

Notes

References

*cite journal
author = Bender, E. A.; Richmond, L. B.; Wormald, N. C.
title = Almost all chordal graphs split
journal = J. Austral. Math. Soc., Series A
volume = 38
year = 1985
issue = 2
id = MathSciNet | id = 0770128
pages = 214–221

*cite book
author = Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy
title = Graph Classes: A Survey
publisher = SIAM Monographs on Discrete Mathematics and Applications
year = 1999
id = ISBN 0-89871-432-X

*cite journal
author = Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin
title = The strong perfect graph theorem
journal = Annals of Mathematics
volume = 164
year = 2006
issue = 1
pages = 51–229
url = http://projecteuclid.org/getRecord?id=euclid.annm/1157656789
id = MathSciNet | id = 2233847

*cite conference
author = Földes, Stéphane; Hammer, Peter L.
title = Split graphs
booktitle = Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977)
pages = 311–315
publisher = Congressus Numerantium, No. XIX, Utilitas Math.
location = Winnipeg
date = 1977a
id = MathSciNet | id = 0505860

*cite journal
author = Földes, Stéphane; Hammer, Peter L.
title = Split graphs having Dilworth number two
journal = Canad. J. Math.
volume = 29
year = 1977b
issue = 3
pages = 666–672
id = MathSciNet | id = 0463041
*citation
last = Golumbic | first = Martin Charles | authorlink = Martin Charles Golumbic
title = Algorithmic Graph Theory and Perfect Graphs
publisher = Academic Press
year = 1980
isbn = 0-12-289260-7.

*cite journal
author = Hammer, Peter L.; Simeone, Bruno
title = The splittance of a graph
journal = Combinatorica
volume = 1
year = 1981
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pages = 275–284
id = MathSciNet | id = 0637832
doi = 10.1007/BF02579333

*cite journal
author = McMorris, F. R.; Shier, D. R.
title = Representing chordal graphs on "K"_1,"n"
journal = Comment. Math. Univ. Carolin.
volume = 24
year = 1983
pages = 489–494

*cite journal
author = Merris, Russell
title = Split graphs
journal = European J. Combin.
volume = 24
year = 2003
issue = 4
pages = 413–430
doi = 10.1016/S0195-6698(03)00030-1
id = MathSciNet | id = 1975945

*cite journal
author = Royle, Gordon F.
title = Counting set covers and split graphs
journal = Journal of Integer Sequences
volume = 3
year = 2000
issue = 2
pages = 00.2.6
id = MathSciNet | id = 1778996
url = http://www.emis.ams.org/journals/JIS/VOL3/ROYLE/royle.pdf

*cite journal
author = Tyshkevich, R. I.
title = The canonical decomposition of a graph
format = In Russian
journal = Doklady Akad. Nauk BSSR
volume = 24
year = 1980
pages = 677–679

*cite journal
author = Tyshkevich, R. I.; Chernyak, A. A.
title = Canonical partition of a graph defined by the degrees of its vertices
format = In Russian
year = 1979
journal = Isv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk
volume = 5
pages = 14–26

*cite journal
author = Tyshkevich, R. I.; Chernyak, A. A.
title = One more method of enumeration of unlabeled combinatorial objects
format = In Russian
year = 1990
journal = Matem. Zametki
volume = 48
pages = 98–105

*cite journal
author = Tyshkevich, R. I.; Melnikow, O. I.; Kotov, V. M.
title = On graphs and degree sequences: the canonical decomposition
format = In Russian
journal = Kibernetica
volume = 6
year = 1981
pages = 5–8

*cite journal
author = Voss, H.-J.
title = Note on a paper of McMorris and Shier
journal = Comment. Math. Univ. Carolin.
volume = 26
year = 1985
pages = 319–322