We consider the following generalization of split graphs: A graph is said to be a (k, l)-graph if its vertex set can be partitioned into k independent sets and l cliques. (Split graphs are obtained by setting k = l = 1). Much of the appeal of split graphs is due to the fact that they are chordal, a property not shared by (k, l)-graphs in general. (For instance, being a (k, 0)-graph is equivalen...

Two graphs G and H with the same vertex set V are P4-isomorphic if every four vertices {a, b, c, d} ⊆ V induce a chordless path (denoted by P4) in G if and only if they induce a P4 in H. We call a graph split-perfect if it is P4-isomorphic to a split graph (i.e., a graph being partitionable into a clique and a stable set). This paper characterizes the new class of split-perfect graphs using the...

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 this paper, we survey results on the hamiltonian and classification problems for split graphs G with the minimum degree δ(G) ≥ |I | − 4. 2000 Mathematics Subject Classification: 05C45, 05C75.

We show that the problem of computing a minimum distortion embedding of a given graph into a path remains NP-hard when the input graph is restricted to a bipartite, cobipartite, or split graph. This implies the NP-hardness of the problem also on chordal, cocomparability, and AT-free graphs. This problem is hard to approximate within a constant factor on arbitrary graphs. We give polynomial-time...

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...

The square of a graph G, denoted 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 chordal graph or even a split graph. We present a polynomial time algorithm that decides whether a given graph...

The isotropic matroid M [IAS(G)] of a graph G is a binary matroid, which is equivalent to the isotropic system introduced by Bouchet. In this paper we discuss four notions of connectivity related to isotropic matroids and isotropic systems. We show that the isotropic system connectivity defined by Bouchet is equivalent to vertical connectivity of M [IAS(G)], and if G has at least four vertices,...

In this paper we investigate how graph problems that are NP-hard in general, but polynomially solvable on split graphs, behave on input graphs that are close to being split. For this purpose we define split+ke and split+kv graphs to be the graphs that can be made split by removing at most k edges and at most k vertices, respectively. We show that problems like treewidth and minimum fill-in are ...

Finding the pebbling number of a graph is harder than NP-complete (Π 2 complete, to be precise). However, for many families of graphs there are formulas or polynomial algorithms for computing pebbling numbers; for example, complete graphs, products of paths (including cubes), trees, cycles, diameter two graphs, and more. Moreover, graphs having minimum pebbling number are called Class 0, and ma...

The Colin de Verdière parameters, μ and ν, are defined to be the maximum nullity of certain real symmetric matrices associated with a given graph. In this work, both of these parameters are calculated for all chordal graphs. For ν the calculation is based solely on maximal cliques, while for μ the calculation depends on split subgraphs. For the case of μ our work extends some recent work on com...