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...

A graph G is called a split graph if the vertex-set V of G can be partitioned into two subsets V1 and V2 such that the subgraphs of G induced by V1 and V2 are empty and complete, respectively. In this paper, we characterize hamiltonian graphs in the class of split graphs with minimum degree δ at least |V1| − 2.

In the Split to Block Vertex Deletion and Threshold problems input is a split graph G an integer k, goal decide whether there set S of vertices size at most k such that G?S block threshold graph, respectively. this paper we give algorithms for these whose running times are O?(2.076k) O?(1.619k),

In this paper we introduce a generalization of the c-planarity testing problem for clustered graphs. Namely, given a clustered graph, the goal of the SPLIT-C-PLANARITY problem is to split as few clusters as possible in order to make the graph c-planar. Determining whether zero splits are enough coincides with testing c-planarity. We show that SPLIT-C-PLANARITY is NP-complete for c-connected clu...

The square of a graph G, denoted by G, is obtained from G by putting an edge between two distinct vertices whenever their distance is two. Then G is called a square root of G. Deciding whether a given graph has a square root is known to be NP-complete, even if the root is required to be a split graph, that is, a graph in which the vertex set can be partitioned into a stable set and a clique. We...

A graph is König-Egerváry if the size of a minimum vertex cover equals the size of a maximum matching in the graph. We show that the problem of deleting at most k vertices to make a given graph König-Egerváry is fixedparameter tractable with respect to k. This is proved using interesting structural theorems on matchings and vertex covers which could be useful in other contexts. We also show an ...

The square of a graph G, denoted by G, is obtained from G by putting an edge between two distinct vertices whenever their distance is two. Then G is called a square root of G. Deciding whether a given graph has a square root is known to be NP-complete, even if the root is required to be a split graph, that is, a graph in which the vertex set can be partitioned into a stable set and a clique. We...

We show how to use split decomposition to compute the weighted clique number and the chromatic number of a graph and we apply these results to some classes of graphs. In particular we present an O(nm) algorithm to compute the chromatic number for all those graphs having a split decomposition in which every prime graph is an induced subgraph of either a Ck or a Ck for some k ≥ 3.

A graph G is called CIS if each maximal clique intersects each maximal stable set in G, and is called almost CIS if it has a unique disjoint pair (C, S) consisting of a maximal clique C and a maximal stable set S. While it is still unknown if there exists a good structural characterization of all CIS graphs, in this note we prove the following Andrade-BorosGurvich conjecture: A graph is almost ...