An almost complement of a graph is obtained from the complement 
by removing the edges of a perfect matching. A graph is called 
almost self-complementary if it is isomorphic to one of its almost 
complements. Thus, an almost self-complementary graph 
is equivalent to a decomposition of the "cocktail party graph"
(i.e., the complete graph minus a 1-factor) into two isomorphic graphs.

Almost self-complementary circulant graphs were first studied by
Dobson and myself (2004). In this presentation I shall describe some
of the properties and constructions of general almost
self-complementary graphs. In particular, I shall give necessary and
sufficient conditions on the order of an almost self-complementary
regular graph, and construct infinite families of almost
self-complementary regular graphs, almost self-complementary
vertex-transitive graphs, and non-cyclically almost
self-complementary circulant graphs.