Rank-width is a graph complexity measure that has many structural properties. It is known that the rank-width of an undirected graph is the maximum over all induced prime graphs with respect to split decomposition and an undirected graph has rank-width at most 1 if and only if it is a distance-hereditary graph. We are interested in an extension of these results to directed graphs. We give sever...

The Classification Problem is the problem of deciding whether a simple graph has chromatic index equals to ∆ or ∆+1, where ∆ is the maximum degree of the graph. It is known that to decide if a graph has chromatic index equals to ∆ is NP-complete. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. The chromatic indexes for some subsets of split graphs, s...

A dominating set D ⊆ V is a split dominating set of a graph G = (V,E) if the induced subgraph of 〈V −D〉 is disconnected. The split domination number γs(G) is the minimum cardinality of a split dominating set of a graph G. In this article, we establish some results on split domination number of Pm⊕ Pn, Pm ⊕ Cn and Cm ⊕ Cn.

In the main this paper introduces the concept of chromatic harmonic polynomials denoted, $H^chi(G,x)$ and chromatic harmonic indices denoted, $H^chi(G)$ of a graph $G$. The new concept is then applied to finding explicit formula for the minimum (maximum) chromatic harmonic polynomials and the minimum (maximum) chromatic harmonic index of certain graphs. It is also applied to split graphs and ce...

A graph G is a chordal-k-generalized split graph if G is chordal and there is a clique Q in G such that every connected component in G[V \ Q] has at most k vertices. Thus, chordal-1-generalized split graphs are exactly the split graphs. We characterize chordal-k-generalized split graphs by forbidden induced subgraphs. Moreover, we characterize a very special case of chordal-2-generalized split ...

Wediscuss various questions around partitioning a split graph into connected parts. Our main result is a polynomial time algorithm that decides whether a given split graph is fully decomposable, that is, whether it can be partitioned into connected parts of orders α1, α2, . . . , αk for every α1, α2, . . . , αk summing up to the order of the graph. In contrast, we show that the decision problem...

The edit operation that contracts edges, which is a fundamental operation in the theory of graph minors, has recently gained substantial scientific attention from the viewpoint of Parameterized Complexity. In this paper, we examine an important family of graphs, namely the family of split graphs, which in the context of edge contractions, is proven to be significantly less obedient than one mig...

A graph G = (V, E) is called a split graph if there exists a partition V = I∪K such that the subgraphs G[I] andG[K] of G induced by I andK 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. Further, a...

A Set S of vertices in a graph G(V,E) is called dominating set,if every vertex v V id either an element or adjacent to S. vertice Total set if S, total G with no isolated subset the such that In this paper we will find perfect Interval graph. We also out induced subgraph regarding which were splitted.