We give a linear-time algorithm to compute the cutwidth of threshold graphs, thereby resolving the computational complexity of cutwidth on this graph class. Although our algorithm is simple and intuitive, its correctness proof relies on a series of non-trivial structural results, and turns out to be surprisingly complex. Threshold graphs are a well-studied subclass of interval graphs and of spl...

We show that the class of perfect graphs known as split graphs can be efficiently recognized in a parallel process environment. Given an undirected graph G = (V, E) with n vertices we define a subgraph G, = (AV, AE) and we prove some properties on G, leading to a constant-time parallel split graph recognition algorithm.

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.

We consider the following vertex-partition problem on graphs, known as the CLUSTER DELETION (CD) problem: given a graph with real nonnegative edge weights, partition the vertices into clusters (in this case, cliques) to minimize the total weight of edges outside the clusters. The decision version of this optimization problem is known to be NP-complete even for unweighted graphs and has been stu...

We consider the following vertex-partition problem on graphs: given a graph with real nonnegative edge weights, partition the vertices into clusters (i.e. cliques) to minimize the total weight of edges out of the clusters. This optimization problem is known to be an NP-complete problem even for unweighted graphs and has been studied extensively in the scope of fixed-parameter tractability (FPT)...

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 study the parameterized complexity of Split Contraction andThreshold Contraction. In these problems we are given a graph G and an integer k and asked whether G can be modified into a split graph or a threshold graph, respectively, by contracting at most k edges. We present an FPT algorithm for Split Contraction, and prove that Threshold Contraction on split graphs, i.e., contracting an input...

A graph is H-free if it has no induced subgraph isomorphic to H . We continue a study into the boundedness of clique-width of subclasses of perfect graphs. We identify five new classes of H-free split graphs whose clique-width is bounded. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions ...