Split graph
In graph theory, a branch of mathematics, 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 (1977a, 1977b), and independently introduced by Tyshkevich and Chernyak (1979).
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:
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 forbidden 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).
Relation to other graph families
From the definition, split graphs are clearly closed under complementation. Another characterization of split graphs involves complementation: they are chordal graphs the complements of which are also chordal. Just as chordal graphs are the intersection graphs of subtrees of trees, split graphs are the intersection graphs of distinct substars of star graphs.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.
