graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, a branch of mathematics, a split graph is a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by , and independently introduced by , where they called these graphs "polar graphs" (). A split graph may have more than one partition into a clique and an independent set; for instance, the

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is a split graph, the vertices of which can be partitioned in three different ways: #the clique and the independent set #the clique and the independent set #the clique and the independent set Split graphs can be characterized in terms of their forbidden induced subgraphs: a graph is split if and only if no

induced subgraph In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and ''all'' of the edges, from the original graph, connecting pairs of vertices in that subset. Definition Formally, let G=(V,E) ...

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 In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...

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. 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 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 In graph theory and order theory, a comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a partial order. Comparability graphs have also been called transitively orientable graphs, partial ...

, 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. The split cographs are exactly the threshold graphs. The split permutation graphs are exactly the interval graphs that have interval graph complements; these are the permutation graphs of skew-merged permutations. Split graphs have cochromatic number 2.

Algorithmic problems

Let be a split graph, partitioned into a clique and an independent set . Then every maximal clique in a split graph is either itself, or the

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of a vertex in . 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: # There exists a vertex in such that is complete. In this case, is a maximum clique and is a maximum independent set. # There exists a vertex in such that is independent. In this case, is a maximum independent set and is a maximum clique. # is a maximal clique and is a maximal independent set. In this case, has a unique partition into a clique and an independent set, is the maximum clique, and is the maximum independent set. Some other optimization problems that are

NP-complete In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any ...

on more general graph families, including

graph coloring In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a Graph (discrete mathematics), graph. The assignment is subject to certain constraints, such as that no two adjacent elements have th ...

, are similarly straightforward on split graphs. Finding a

Hamiltonian cycle In the mathematics, mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path (graph theory), path in an undirected or directed graph that visits each vertex (graph theory), vertex exactly once. A Hamiltonian cycle (or ...

remains

even for split graphs which are strongly chordal graphs. It is also well known that the Minimum Dominating Set problem remains

for split graphs.

Degree sequences

One remarkable property of split graphs is that they can be recognized solely from their degree sequences. Let the degree sequence of a graph be , and let be the largest value of such that . Then is a split graph if and only if :

\sum_^m d_i = m(m-1) + \sum_^n d_i.

If this is the case, then the vertices with the largest degrees form a maximum clique in , and the remaining vertices constitute an independent set.; ; ; , Theorem 6.7 and Corollary 6.8, p. 154; , Theorem 13.3.2, p. 203. further investigates the degree sequences of split graphs. The splittance of an arbitrary graph measures the extent to which this inequality fails to be true. If a graph is not a split graph, then the smallest sequence of edge insertions and removals that make it into a split graph can be obtained by adding all missing edges between the vertices with the largest degrees, and removing all edges between pairs of the remaining vertices; the splittance counts the number of operations in this sequence.

Counting split graphs

showed that ( unlabeled) ''n''-vertex split graphs are in

one-to-one correspondence In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivale ...

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 :1, 2, 4, 9, 21, 56, 164, 557, 2223, 10766, 64956, 501696, ... . This enumerative result was also proved earlier by .

Notes

References