Split graph
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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).[1]
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).[2]
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[edit] Relation to other graph families
From the definition, split graphs are clearly closed under complementation.[3] Another characterization of split graphs involves complementation: they are chordal graphs the complements of which are also chordal.[4] Just as chordal graphs are the intersection graphs of subtrees of trees, split graphs are the intersection graphs of distinct substars of stars.[5] 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.[6] The split cographs are exactly the threshold graphs, and the split permutation graphs are exactly the interval graphs that have interval graph complements.[7] Split graphs have cochromatic number 2.
[edit] 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:[8]
- 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.
[edit] 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 d1 ≥ d2 ≥ ... ≥ dn, and let m be the largest value of i such that di ≥ i - 1. Then G is a split graph if and only if
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.[9]
[edit] 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, ... (sequence A048194 in OEIS). This result was independently proved earlier by Tyshkevich and Chernyak in 1990 (in Russian).
[edit] Notes
- ^ 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.
- ^ Földes and Hammer (1977a); Golumbic (1980), Theorem 6.3, p. 151.
- ^ Golumbic (1980), Theorem 6.1, p. 150.
- ^ Földes and Hammer (1977a); Golumbic (1980), Theorem 6.3, p. 151; Brandstädt et al. (1999), Theorem 3.2.3, p. 41.
- ^ McMorris and Shier (1983); Voss (1985); Brandstädt et al. (1999), Theorem 4.4.2, p. 52.
- ^ Földes and Hammer (1977b); Golumbic (1980), Theorem 9.7, page 212.
- ^ Brandstädt et al. (1999), Corollary 7.1.1, p. 106, and Theorem 7.1.2, p. 107.
- ^ Hammer and Simeone (1981); Golumbic (1980), Theorem 6.2, p. 150.
- ^ 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.
[edit] References
- Bender, E. A.; Richmond, L. B.; Wormald, N. C. (1985). "Almost all chordal graphs split". J. Austral. Math. Soc., Series A 38 (2): 214–221. MR0770128.
- Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999). Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications. ISBN 0-89871-432-X.
- Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2006). "The strong perfect graph theorem". Annals of Mathematics 164 (1): 51–229. MR2233847.
- Földes, Stéphane; Hammer, Peter L. (1977a). "Split graphs". Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977): 311–315, Winnipeg: Congressus Numerantium, No. XIX, Utilitas Math.. MR0505860.
- Földes, Stéphane; Hammer, Peter L. (1977b). "Split graphs having Dilworth number two". Canad. J. Math. 29 (3): 666–672. MR0463041.
- Golumbic, Martin Charles (1980), Algorithmic Graph Theory and Perfect Graphs, Academic Press, ISBN 0-12-289260-7.
- Hammer, Peter L.; Simeone, Bruno (1981). "The splittance of a graph". Combinatorica 1 (3): 275–284. doi:. MR0637832.
- McMorris, F. R.; Shier, D. R. (1983). "Representing chordal graphs on K1,n". Comment. Math. Univ. Carolin. 24: 489–494.
- Royle, Gordon F. (2000). "Counting set covers and split graphs". Journal of Integer Sequences 3 (2): 00.2.6. MR1778996.
- Tyshkevich, R. I. (1980). "The canonical decomposition of a graph" (In Russian). Doklady Akad. Nauk BSSR 24: 677–679.
- Tyshkevich, R. I.; Chernyak, A. A. (1979). "Canonical partition of a graph defined by the degrees of its vertices" (In Russian). Isv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk 5: 14–26.
- Tyshkevich, R. I.; Chernyak, A. A. (1990). "One more method of enumeration of unlabeled combinatorial objects" (In Russian). Matem. Zametki 48: 98–105.
- Tyshkevich, R. I.; Melnikow, O. I.; Kotov, V. M. (1981). "On graphs and degree sequences: the canonical decomposition" (In Russian). Kibernetica 6: 5–8.
- Voss, H.-J. (1985). "Note on a paper of McMorris and Shier". Comment. Math. Univ. Carolin. 26: 319–322.
