We study the problem of adding an inclusion minimal set of edges to a given arbitrary graph so that the resulting graph is a split graph, called a minimal split completion of the input graph. Minimal completions of arbitrary graphs into chordal and interval graphs have been studied previously, and new results have been added recently. We extend these previous results to split graphs by giving a...

We study the parameterized complexity of Split Contraction andThreshold Contraction. In these problems we are given a graph G and an integer k and asked whether G can be modified into a split graph or a threshold graph, respectively, by contracting at most k edges. We present an FPT algorithm for Split Contraction, and prove that Threshold Contraction on split graphs, i.e., contracting an input...

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

A graph G = (V, E) is called a split graph if there exists a partition V = I ∪K such that the subgraphs G[I ] and G[K] of G induced by I and K are empty and complete graphs, respectively. In 1980, Burkard and Hammer gave a necessary condition for a split graph G with |I | < |K| to be hamiltonian. We will call a split graph G with |I | < |K| satisfying this condition a Burkard-Hammer graph. Furt...

Let Γ denote a Q-polynomial distance-regular graph with diameter at least three and standard module V . We introduce two direct sum decompositions of V . We call these the displacement decomposition for Γ and the split decomposition for Γ. We describe how these decompositions are related.

It is proved that a split graph is an absolute retract of split graphs if and only if a partition of its vertex set into a stable set and a complete set is unique or it is a complete split graph. Three equivalent conditions for a split graph to be an absolute retract of the class of all graphs are given. It is finally shown that a reflexive split graph G is an absolute retract of reflexive spli...

We present an algorithm that supports operations for modifying a split graph by adding edges or vertices and deleting edges, such that after eachmodification the graph is repaired to become a split graph in a minimal way. In particular, if the graph is not split after the modification, the algorithm computes a minimal, or if desired even a minimum, split completion or deletion of themodified gr...

An edge-coloring of a graph is an assignment of colors to its edges such that no adjacent edges have the same color. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. Split graphs have been introduced by Földes and Hammer [4] and it is a well-studied class of graphs. However, the problem of deciding the chromatic index of any split graph remains unsolv...

A linear-time algorithm for decomposing a graph into split components is presented. The algorithm uses a new graph transformation technique to gradually transform the given graph so that every split component in it is transformed into a subgraph with very simple structure which can be easily identified. Once the split components are determined, the triconnected components of the graph are easil...