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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)

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)

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 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)