Final answer:
In the complement graph G' of a disconnected graph G, every vertex is connected to all other vertices in different components, which proves the diameter of G' is at most 2 because any two vertices can be connected directly or through a single intermediary vertex.
Step-by-step explanation:
The question pertains to the proof of the property of the complement of a disconnected graph (denoted as G'). The diameter of a graph is defined as the longest shortest path between any two vertices in the graph. In the case of a disconnected graph, there are at least two vertices that are not connected by a path, and therefore, the classic notion of diameter is infinite or undefined for the graph itself. However, when considering the complement of a disconnected graph, G', every vertex in one disconnected component of G is connected to every vertex in other disconnected components of G' since G' contains all the edges that are not present in G.
To prove that the diameter of G' is at most 2, we note that for any two vertices u and v in G', there are two possibilities. If u and v are in different components of the original disconnected graph G, there is a direct edge between them in G' by the definition of a complement graph. If u and v are in the same component of G, then there must exist at least one vertex w in a different component (since G is disconnected). Both u and w, and v and w are connected directly by an edge in G'. Therefore, we can travel from u to v via w with a path length of 2. Thus, the diameter of G' is at most 2.