An undirected graph G = (V, E) is a probe split graph if its vertex set can be partitioned into two sets, N (nonprobes) and P (probes) where N is independent and there exists E′ ⊆ N ×N such that G′ = (V, E ∪ E′) is a split graph. Recently Chang et al. gave an O(V (V +E)) time recognition algorithm for probe split graphs. In this article we give O(V 2 +V E) time recognition algorithms and charac...

Došlić et al. defined the Mostar index of a graph G as ∑uv∈E(G)|nG(u,v)−nG(v,u)|, where, for an edge uv G, term nG(u,v) denotes number vertices that have smaller distance in to u than v. Contributing conjectures posed by al., we show bipartite graphs order n is at most 318n3, and split 427n3.

We give a linear-time algorithm to compute the cutwidth of threshold graphs, thereby resolving the computational complexity of cutwidth on this graph class. Threshold graphs are a well-studied subclass of interval graphs and of split graphs, both of which are unrelated subclasses of chordal graphs. To complement our result, we show that cutwidth is NPcomplete on split graphs, and consequently a...

A regular covering projection ℘: X̃ → X of connected graphs is G-admissible if G lifts along ℘. Denote by G̃ the lifted group, and let CT(℘) be the group of covering transformations. The projection is called G-split whenever the extension CT(℘) → G̃ → G splits. In this paper, split 2-covers are considered, with a particular emphasis given to cubic symmetric graphs. Supposing that G is transitive o...

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...

The concept of split domination number was introduced by Kulli and Janakiram. Fink Jacobson the notion k-domination in graphs. In this paper, we acquaint with k-split k-non for some zero-divisor graphs ϑ-Obrazom

In this paper, we investigate the well-studied Hamiltonian cycle problem, and present an interesting dichotomy result on split graphs. T. Akiyama, T. Nishizeki, and N. Saito [22] have shown that the Hamiltonian cycle problem is NP-complete in planar bipartite graph with maximum degree 3. Using this reduction, we show that the Hamiltonian cycle problem is NP-complete in split graphs. In particul...

We consider two problems, namely Min Split-coloring and Min Cocoloring, that generalize the classical Min Coloring problem by using not only stable sets but also cliques to cover all the vertices of a given graph. We prove the NP-hardness of some cases. We derive approximation results for Min Split-coloring and Min Cocoloring in line graphs, comparability graphs and general graphs. This provide...

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...