existence of disconnected, self-complementary graph Does there exist a disconnected self-complementary graph?
I think no, because if graph is disconnected(connected), then the complementary will be connected(disconnected). Am I proceeding in the right direction. Thanks for your help.

You are proceeding in the right direction.
However, if the graph is connected, then its complemented may be connected too; think of $K_5$, split in the pentagon and the five-pointed star.

To show that indeed the complement of a disconnected graph is connected, let $u, v$ be vertices of $G$.
If $u,v$ are in different connected components (CCs), then there is an edge between them in $G^c$. If they are in the same CC, then there is a vertex $w$ in a different CC since $G$ is not connected. Both $u$ and $v$ have an edge to $w$ in $G^c$, hence there is a path from $u$ to $v$ in $G^c$.
