Figure 9

Illustration of conditions (1) and (2) 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. (a) and (b) illustrate the case of lca(u,v)≠u (where two rooted subtrees T(gua(u,v)) and T(gua(v,u)) are not isomorphic to each other by copy(gua(v,u))=0) and the case of lca(u,v)=u, respectively (note that lca(u,v)≠v by dfs(u)<dfs(v)). Conditions (1) and (2) exclude any ancestor w of u or v such that copy(w)=1 (otherwise we can prove that there is a lexicographically smaller pair (u^′,v^′) such that T+u^′v^′ is isomorphic to T+u v).