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List partitions generalize list colourings and list homomorphisms. Each symmetric matrix M over 0; 1; deenes a list partition problem. Diierent choices of the matrix M lead to many well-known graph the-oretic problems including the problem of recognizing split graphs and their generalizations, nding homogeneous sets, joins, clique cutsets, stable cutsets,… (More)

We introduce a parametrized family of graph problems that includes several well-known graph partition problems as special cases. We develop tools which allow us to classify the complexity of many problems in this family, and in particular lead us to a complete classiication for small values of the parameters. Along the way, we obtain a variety of speciic… (More)

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… (More)

A skew partition as defined by Chvátal is a partition of the vertex set of a graph into four nonempty parts AA BB CC D such that there are all possible edges between A and B and no edges between C and D. We present a polynomial-time algorithm for testing whether a graph admits a skew partition. Our algorithm solves the more general list skew partition… (More)

A homogeneous set is a non-trivial module of a graph, i.e. a non-empty, non-unitary, proper subset of a graph's vertices such that all its elements present exactly the same outer neighborhood. Given two published an all-fast O(n 2 2) algorithm which was recently proven wrong [5], so that the HSSP's known upper bound would have been reset thereafter at… (More)

It is well known that a clique with k + 1 vertices is the only minimal obstruction to k-colourability of chordal graphs. A similar result is known for the existence of a cover by cliques. Both of these problems are in fact partition problems, restricted to chordal graphs. The first seeks partitions into k independent sets, and the second is equivalent to… (More)

A graph is a 1-join composition if its vertex set can be partitioned into four nonempty sets

We study the concept of an H-partition of the vertex set of a graph G, which includes all vertex partitioning problems into four parts which we require to be nonempty with only external constraints according to the structure of a model graph H, with the exception of two cases, one that has already been classified as polynomial, and the other one remains… (More)

In Hell et al. (2004), we have previously observed that, in a chordal graph G, the maximum number of independent r-cliques (i.e., of vertex disjoint subgraphs of G, each isomorphic to K r , with no edges joining any two of the subgraphs) equals the minimum number of cliques of G that meet all the r-cliques of G. When r = 1, this says that chordal graphs… (More)