In Algorithmic Archipelago, there are two islands that each have several towns connected by roads. These towns and roads form a connected graph on each island, labelled G$1$

and G$2$

respectively. A bridge is going to be built between two towns, one from each island, effectively introducing a new edge between one vertex in G$1$

and one vertex in G$2$

$.$ This new graph represented the joined islands will be called H

$.$