Concept guide
Trees: deciding that a 6-vertex graph is a tree, with its leaves, a unique path and the unique cycle after adding an edge
Let G=(V,E) with V={1,2,3,4,5,6} and E={{1,2},{2,3},{2,4},{4,5},{4,6}}. (1) Prove from the MKO's definition that G is a tree (V nonempty, connected, no simple cycle); (2) list its leaves; (3) find the path from 3 to 6 and prove it is unique; (4) add the edge {3,6}, write the simple cycle it creates and prove it is the only simple cycle; (5) contrast: a triangle plus an isolated vertex has |E|=|V|−1 but is not a tree; a single vertex without edges is a tree with no leaf.