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| [[File:Split graph.svg|thumb|240px|A split graph, partitioned into a clique and an independent set.]]
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| In [[graph theory]], a branch of mathematics, a '''split graph''' is a graph in which the vertices can be partitioned into a [[clique (graph theory)|clique]] and an [[Independent set (graph theory)|independent set]]. Split graphs were first studied by {{harvs|last1=Földes|author2-link=Peter Hammer|last2=Hammer|year=1977a|year2=1977b|txt}}, and independently introduced by {{harvs|author1-link=Regina Tyshkevich|last1=Tyshkevich|last2=Chernyak|year=1979|txt}}.<ref>{{harvtxt|Földes|Hammer|1977a}} had a more general definition, in which the graphs they called split graphs also included [[bipartite graph]]s (that is, graphs that be partitioned into two independent sets) and the [[complement (graph theory)|complements]] of bipartite graphs (that is, graphs that can be partitioned into two cliques). {{harvtxt|Földes|Hammer|1977b}} use the definition given here, which has been followed in the subsequent literature; for instance, it is {{harvtxt|Brandstädt|Le|Spinrad|1999}}, Definition 3.2.3, p.41.</ref>
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| A split graph may have more than one partition into a clique and an independent set; for instance, the path ''a''–''b''–''c'' is a split graph, the vertices of which can be partitioned in three different ways:
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| #the clique {''a'',''b''} and the independent set {''c''}
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| #the clique {''b'',''c''} and the independent set {''a''}
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| #the clique {''b''} and the independent set {''a'',''c''}
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| Split graphs can be characterized in terms of their [[forbidden induced subgraph]]s: a graph is split if and only if no [[induced subgraph]] is a [[cycle graph|cycle]] on four or five vertices, or a pair of disjoint edges (the complement of a 4-cycle).<ref>{{harvtxt|Földes|Hammer|1977a}}; {{harvtxt|Golumbic|1980}}, Theorem 6.3, p. 151.</ref>
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| ==Relation to other graph families==
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| From the definition, split graphs are clearly closed under [[complement (graph theory)|complementation]].<ref>{{harvtxt|Golumbic|1980}}, Theorem 6.1, p. 150.</ref> Another characterization of split graphs involves complementation: they are [[chordal graph]]s the [[complement (graph theory)|complements]] of which are also chordal.<ref>{{harvtxt|Földes|Hammer|1977a}}; {{harvtxt|Golumbic|1980}}, Theorem 6.3, p. 151; {{harvtxt|Brandstädt|Le|Spinrad|1999}}, Theorem 3.2.3, p. 41.</ref> Just as chordal graphs are the [[intersection graph]]s of subtrees of trees, split graphs are the intersection graphs of distinct substars of [[star graph]]s.<ref>{{harvtxt|McMorris|Shier|1983}}; {{harvtxt|Voss|1985}}; {{harvtxt|Brandstädt|Le|Spinrad|1999}}, Theorem 4.4.2, p. 52.</ref> [[Almost all]] chordal graphs are split graphs; that is, in the limit as ''n'' goes to infinity, the fraction of ''n''-vertex chordal graphs that are split approaches one.<ref>{{harvtxt|Bender|Richmond|Wormald|1985}}.</ref>
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| Because chordal graphs are [[perfect graph|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 {{harvtxt|Chudnovsky|Robertson|Seymour|Thomas|2006}} of the [[Strong Perfect Graph Theorem]].
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| 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.<ref>{{harvtxt|Földes|Hammer|1977b}}; {{harvtxt|Golumbic|1980}}, Theorem 9.7, page 212.</ref> The split [[cograph]]s are exactly the [[threshold graph]]s, and the split [[permutation graph]]s are exactly the interval graphs that have interval graph complements.<ref>{{harvtxt|Brandstädt|Le|Spinrad|1999}}, Corollary 7.1.1, p. 106, and Theorem 7.1.2, p. 107.</ref> Split graphs have [[cocoloring|cochromatic number]] 2.
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| ==Maximum clique and maximum independent set==
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| 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 (graph theory)|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:<ref>{{harvtxt|Hammer|Simeone|1981}}; {{harvtxt|Golumbic|1980}}, Theorem 6.2, p. 150.</ref>
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| # 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.
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| # 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.
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| # ''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.
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| Some other optimization problems that are [[NP-complete]] on more general graph families, including [[graph coloring]], are similarly straightforward on split graphs.
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| ==Degree sequences==
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| One remarkable property of split graphs is that they can be recognized solely from their [[degree sequence]]s. Let the degree sequence of a graph ''G'' be ''d''<sub>1</sub> ≥ ''d''<sub>2</sub> ≥ ... ≥ ''d''<sub>''n''</sub>, and let ''m'' be the largest value of ''i'' such that ''d''<sub>''i''</sub> ≥ ''i'' - 1. Then ''G'' is a split graph if and only if
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| :<math>\sum_{i=1}^m d_i = m(m-1) + \sum_{i=m+1}^n d_i.</math>
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| If this is the case, then the ''m'' vertices with the largest degrees form a maximum clique in ''G'', and the remaining vertices constitute an independent set.<ref>{{harvtxt|Hammer|Simeone|1981}}; {{harvtxt|Tyshkevich|1980}}; {{harvtxt|Tyshkevich|Melnikow|Kotov|1981}}; {{harvtxt|Golumbic|1980}}, Theorem 6.7 and Corollary 6.8, p. 154; {{harvtxt|Brandstädt|Le|Spinrad|1999}}, Theorem 13.3.2, p. 203. {{harvtxt|Merris|2003}} further investigates the degree sequences of split graphs.</ref>
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| ==Counting split graphs==
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| {{harvtxt|Royle|2000}} showed that ''n''-vertex split graphs with ''n'' are in [[one-to-one correspondence]] with certain [[Sperner families]]. Using this fact, he determined a formula for the number of (nonisomorphic) split graphs on ''n'' vertices. For small values of ''n'', starting from ''n'' = 1, these numbers are
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| :1, 2, 4, 9, 21, 56, 164, 557, 2223, 10766, 64956, 501696, ... {{OEIS|id = A048194}}. | |
| This [[graph enumeration]] result was also proved earlier by {{harvtxt|Tyshkevich|Chernyak|1990}}.
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| ==Notes==
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| {{reflist|2}}
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| {{refend}}
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| ==Further reading==
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| *A chapter on split graphs appears in the book by [[Martin Charles Golumbic]], "Algorithmic Graph Theory and Perfect Graphs".
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| [[Category:Graph families]]
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| [[Category:Intersection classes of graphs]]
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| [[Category:Perfect graphs]]
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