Cographs

Abstract

Cographs--defined most simply as complete graphs with colored lines--both dualize and generalize ordinary graphs, and promise a comparably wide range of applications. This article introduces them by examples, catalogues, and elementary properties. Any finite cograph may be realized in several ways, including inner products, polynomials, geometrically, or by "fat intersections." Particular classes then considered include sum cographs (points in Z or Zn; the line C(P,Q) joining points P and Q defined C(P,Q) = P+Q); difference cographs (C(P,Q) = |P-Q|; and intersection cographs (points are sets; C(P,Q) = P intersect Q). Intersection cographs, especially, promise many applications; described here are some to aesthetics. Point-line cographs turn out equivalent to linear spaces. Finally solved here is an interesting group-theoretic problem arising from group cographs (points in a group; C(P,Q) = {PQ,QP}).