Illustration of condition (3) for admissible pairs. The black dotted line joins two vertices u and v, which will be the edge to create a cycle in the 1-augmented tree T+u v. Condition (1) and (2) exclude any ancestor w of u or v such that copy(w)=1 except for w=gua(v,u). The assumption copy(gua(v,u))=1 requires the rooted subtree Tgua(u,v) to be isomorphic to Tgua(v,u) and condition (3)-(i) requires Tgua(u,v) to be located immediately on the left of Tgua(v,u) among the subtrees rooted at the children of lca(u,v). Then Tgua(v,u) contains a copy of u (i.e., ). Similarly Tgua(u,v) contains a copy of v (i.e., ). Condition (3)-(ii) requires (otherwise we can prove that is a lexicographically smaller pair such that is isomorphic to T+u v). Although Tgua(u,v) and Tgua(v,u) are isomorphic to each other, only paths and nodes relevant for explanation are shown in this figure.