Closure Concepts: A Survey

In this paper we survey results of the following type (known as closure results). Let P be a graph property, and let C…u; v† be a condition on two nonadjacent vertices u and v of a graph G. Then G ‡ uv has property P if and only if G has property P. The ®rst and now well-known result of this type was established by Bondy and ChvaÂtal in a paper published in 1976: If u and v are two nonadjacent vertices with degree sum n in a graph G on n vertices, then G ‡ uv is hamiltonian if and only if G is hamiltonian. Based on this result, they de®ned the n-closure cln…G† of a graph G on n vertices as the graph obtained from G by recursively joining pairs of nonadjacent vertices with degree sum n until no such pair remains. They showed that cln…G† is well-de®ned, and that G is hamiltonian if and only if cln…G† is hamiltonian. Moreover, they showed that cln…G† can be obtained by a polynomial algorithm, and that a Hamilton cycle in cln…G† can be transformed into a Hamilton cycle of G by a polynomial algorithm. As a consequence, for any graph G with cln…G† ˆ Kn (and nV 3), a Hamilton cycle can be found in polynomial time, whereas this problem is NP-hard for general graphs. All classic su1⁄2cient degree conditions for hamiltonicity imply a complete n-closure, so the closure result yields a common generalization as well as an easy proof for these conditions. In their ®rst paper on closures, Bondy and ChvaÂtal gave similar closure results based on degree sum conditions for nonadjacent vertices for other graph properties. Inspired by their ®rst results, many authors developed other closure concepts for a variety of graph properties, or used closure techniques as a tool for obtaining deeper su1⁄2ciency results with respect to these properties. Our aim is to survey this progress on closures made in the past (more than) twenty years.