It is shown in this note that it can be recognized in polynomial time whether the vertex set of a finite undirected graph can be partitioned into one or two independent sets and one or two cliques. Such graphs generalize bipartite and split graphs and the result also shows that it can be recognized in polynomial time whether a graph can be partitioned into two split graphs. A time bound O(n) is...

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).

We show that the mixed search number and the linear-width of interval graphs and of split graphs can be computed in linear time and in polynomial time, respectively.

The quadratic embedding constant (QE constant) of a graph is new characteristic value defined through the distance matrix. We derive formulae for QE constants join two regular graphs, double graphs and certain lexicographic product graphs. Examples include complete bipartite wheel friendship completely split graph, some associated to strongly

A graph G is called a ( p, q)-split graph if its vertex set can be partitioned into A, B so that the order of the largest independent set in A is at most p and the order of the largest complete subgraph in B is at most q. Applying a well-known theorem of Erdo s and Rado for 2-systems, it is shown that for fixed p, q, ( p, q)-split graphs can be characterized by excluding a finite set of forbidd...

In this note, we investigate three versions of the overfull property for graphs and their relation to the edge-coloring problem. Each of these properties implies that the graph cannot be edge-colored with colors, where is the maximum degree. The three versions are not equivalent for general graphs. However, we show that some equivalences hold for the classes of indiierence graphs, split graphs,...

The complexity of the simple maxcut problem is investigated for several special classes of graphs. It is shown that this problem is NP-complete when restricted to one of the following classes: chordal graphs, undirected path graphs, split graphs, tripartite graphs, and graphs that are the complement of a bipartite graph. The problem can be solved in polynomial time, when restricted to graphs wi...

This paper solves the Non Linear Schrodinger Equation using the Split Step Fourier method for modeling an optical fiber. The model generates a complex wave of optical pulses and using the results obtained two graphs namely Loss versus Wavelength and Dispersion versus Wavelength are generated. Taking Chromatic Dispersion and Polarization Mode Dispersion losses into account, the graphs generated ...

We introduce a generalization of the well known graph (vertex) coloring problem, which we call the problem of component coloring of graphs. Given a graph, the problem is to color the vertices using the minimum number of colors so that the size of each connected component of the subgraph induced by the vertices of the same color does not exceed C. We give a linear time algorithm for the problem ...