Some results on generalized self-complementary graphs

Abstract

Let [Formula: see text] be a partition of vertex set [Formula: see text] of order [Formula: see text] of a graph [Formula: see text]. The [Formula: see text]-complement of [Formula: see text] denoted by [Formula: see text] is defined as for all [Formula: see text] and [Formula: see text] in [Formula: see text], [Formula: see text], remove the edges between [Formula: see text] and [Formula: see text] in [Formula: see text] and add the edges between [Formula: see text] and [Formula: see text] which are not in [Formula: see text]. The graph [Formula: see text] is called [Formula: see text]-self-complementary if [Formula: see text]. For a graph [Formula: see text], [Formula: see text]-complement of [Formula: see text] denoted by [Formula: see text] is defined as for each [Formula: see text] remove the edges of [Formula: see text] inside [Formula: see text] and add the edges of [Formula: see text] by joining the vertices of [Formula: see text]. The graph [Formula: see text] is called [Formula: see text]-self-complementary if [Formula: see text] for some partition [Formula: see text] of order [Formula: see text]. In this paper, we determine generalized self-complementary graphs of forest, double star and unicyclic graphs.