A graph is unipolar if it can be partitioned into a clique and a disjoint union of cliques, and a graph is a generalised split graph if it or its complement is unipolar. A unipolar partition of a graph can be used to find efficiently the clique number, the stability number, the chromatic number, and to solve other problems that are hard for general graphs. We present an O(n)-time algorithm for ...

We prove that a graph G contains no induced 5-vertex path and no induced complement of a 5-vertex path if and only if G is obtained from 5-cycles and split graphs by repeatedly applying the following operations: substitution, split unification, and split unification in the complement (where split unification is a new class-preserving operation that is introduced in this paper).

Splitting off a pair of edges su; sv in a graph G means replacing these two edges by a new edge uv. This operation is well-known in graph theory. Let G = (V + s; E + F ) be a graph which is k-edge-connected in V and suppose that jF j is even. Here F denotes the set of edges incident with s. Lovász [12] proved that if k 2 then the edges in F can be split off in pairs preserving the k-edge-connec...

A graph G = (V,E) is a unipolar graph if there exits a partition V = V1 ∪ V2 such that, V1 is a clique and V2 induces the disjoint union of cliques. The complement-closed class of generalized split graphs are those graphs G such that either G or the complement of G is unipolar. Generalized split graphs are a large subclass of perfect graphs. In fact, it has been shown that almost all C5-free (a...

The concept of connectedness plays an important role in many networks. Digraphs are considered as an excellent modeling tool and are used to model many types of relations amongst any physical situations. In this paper the concept of strong non-split domination in directed graph D has been introduced by considering the dominating set S is a strong non-split dominating set if the complement of S ...

Matrix partition problems generalize a number of natural graph partition problems, and have been studied for several standard graph classes. We prove that each matrix partition problem has only finitely many minimal obstructions for split graphs. Previously such a result was only known for the class of cographs. (In particular, there are matrix partition problems which have infinitely many mini...

The Colin de Verdière parameters, μ and ν, are defined to be the maximum nullity of certain real symmetric matrices associated with a given graph. In this work, both of these parameters are calculated for all chordal graphs. For ν the calculation is based solely on maximal cliques, while for μ the calculation depends on split subgraphs. For the case of μ our work extends some recent work on com...

We prove that a connected graph has linear rank-width 1 if and only if it is a distance-hereditary graph and its split decomposition tree is a path. An immediate consequence is that one can decide in linear time whether a graph has linear rank-width at most 1, and give an obstruction if not. Other immediate consequences are several characterisations of graphs of linear rankwidth 1. In particula...

Split procedures have been proved to be efficient within global framework optimization for routing problems by splitting giant tour into trips. This is done by generating optimal shortest path within an auxiliary graph built from the giant tour. An efficient application has been introduced for the first time by Lacomme et al. (2001) within a metaheuristic approach to solve the Capacitated Arc R...