Skew partition

In graph theory, a skew partition of a graph is a partition of its vertices into two subsets, such that the induced subgraph formed by one of the two subsets is disconnected and the induced subgraph formed by the other subset is the complement of a disconnected graph. Skew partitions play an important role in the theory of perfect graphs. A skew partition of a graph is a partition of its vertices into two subsets and for which the induced subgraph is disconnected and the induced subgraph is the complement of a disconnected graph (co-disconnected). Equivalently, a skew partition of a graph may be described by a partition of the vertices of into four subsets , , , and , such that there are no edges from to and such that all possible edges from to exist; for such a partition, the induced subgraphs and are disconnected and co-disconnected respectively, so we may take and . Every path graph with four or more vertices has a skew partition, in which the co-disconnected set is one of the interior edges of the path and the disconnected set consists of the vertices on either side of this edge. However, it is not possible for a cycle graph of any length to have a skew partition: no matter which subsets of the cycle are chosen as the set , the complementary set will have the same number of connected components, so it is not possible for to be disconnected and to be co-disconnected. If a graph has a skew partition, so does its complement. For instance, the complements of path graphs have skew partitions, and the complements of cycle graphs do not. If a graph is itself disconnected, then with only three simple exceptions (an empty graph, a graph with one edge and three vertices, or a four-vertex perfect matching) it has a skew partition, in which the co-disconnected side of the partition consists of the endpoints of a single edge and the disconnected side consists of all other vertices. For the same reason, if the complement of a graph is disconnected, then with a corresponding set of three exceptions it must have a skew partition.