An alternative way of representing a partition by means of a graph is to use p labelled edges instead of labelled vertices. In this form, the graph of ϒ* =

{υ1,…,υν} comprises p labelled edges emanating from α unlabelled vertices, giving a total of α+p vertices of which p are ‘free’. If ϒ = {υ1

,…,υν} is another partition of the same indices represented as a graph in the same way, we define the graph ϒ ⊗ ϒ* by connecting corresponding free vertices of the two graphs. This gives a graph with α + ν unlabelled vertices and p labelled edges, parallel edges being permitted. Show that the graph ϒ ⊕ ϒ* is connected if and only if ϒ ⋁ ϒ* =

1.