To prove that the complement of a disconnected graph G is connected, the solution shows that for any two distinct vertices u and v in the complement graph G-1, there is always a path between them of length at most 2. This is because if u and v are in different components of G, they are adjacent in G-1, and if in the same component, there is a third vertex w in a different component such that u and v are both adjacent to w in G-1.

1. Prove that if G is any disconnected graph then its complement G-1 is a connected graph.
(G-1 is G with a bar over it).
Solution
Since G is not connected, there are at least 2 connected components. We claim that
in the complement graph there is a path of length at most 2 between any 2 distinct vertices. This
proves in particular that the complement is connected. Suppose u, v are distinct vertices. If u, v
are in different connected components of G then there is no u-v edge in G => u and v are
adjacent in the complement. Suppose now that u, v are in the same connected component C of
G. Since G has at least 2 connected components, let C' be another connected component of G.
Then there is no C-C' edge in G which means that in G, for any w in C' there is no u-w edge or
v-w edge => both these occur in the complement graph, so there is a path from v to w of length
2. This proves the claim and completes the proof.