More on almost self-complementary graphs

Abstract

A graph X is called almost self-complementary if it is isomorphic to one of its almost complements X c − I , where X c denotes the complement of X and I a perfect matching (1-factor) in X c . If I is a perfect matching in X c and φ : X → X c − I is an isomorphism, then the graph X is said to be fairly almost self-complementary if φ preserves I setwise, and unfairly almost self-complementary if it does not. In this paper we construct connected graphs of all possible orders that are fairly and unfairly almost self-complementary, fairly but not unfairly almost self-complementary, and unfairly but not fairly almost self-complementary, respectively, as well as regular graphs of all possible orders that are fairly and unfairly almost self-complementary. Two perfect matchings I and J in X c are said to be X -non-isomorphic if no isomorphism from X + I to X + J induces an automorphism of X . We give a constructive proof to show that there exists a graph X that is almost self-complementary with respect to two X -non-isomorphic perfect matchings for every even order greater than or equal to four.