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Proposition. Every graph with at least $2$nodes$,$ and in which all nodes have degree $1,$must be connected.

Proof. Let P$($n$)$be the predicate $$Every graph with exactly n nodes, all of which have degree $1,$is connected.$$We will prove $$n $2$P$($n$)$by induction:

Base Case: Suppose n $=2.$There is only one $2-$node graph in which all nodes have degree $1,$which is the graph that has an edge connecting its two nodes. This graph is connected, and so P$(2)$holds$.$

Inductive Step: Let n $3$be any integer and assume the inductive hypothesis P$($n $1).$Let G be a graph with exactly n $1$nodes that all have degree $1.$By the inductive hypothesis, G must be connected.

Now add any one new node v to G and one or more edges from v to other nodes in the graph. Call the new graph G$.$Note that G$$has exactly n nodes, and they all have degree $1.$Additionally$,$ since G is connected and there is an edge from v to a node in G$,$there is a path from v to all other nodes. So G$$is connected, and so P$($n$)$holds$.$