Figure 10

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 T_gua(u,v) to be isomorphic to T_gua(v,u) and condition (3)-(i) requires T_gua(u,v) to be located immediately on the left of T_gua(v,u) among the subtrees rooted at the children of lca(u,v). Then T_gua(v,u) contains a copy $û$ of u (i.e., $\text{dfs}(û)=\text{dfs}(u)+\left|T_{\text{gua}(u,v)}\right|$). Similarly T_gua(u,v) contains a copy $\widehat{v}$ of v (i.e., $\text{dfs}(\widehat{v})=\text{dfs}(v)-\left|T_{\text{gua}(u,v)}\right|$). Condition (3)-(ii) requires $\text{dfs}(û)\le \text{dfs}(v)$ (otherwise we can prove that $(\widehat{v},û)$ is a lexicographically smaller pair such that $T+\widehat{v}û$ is isomorphic to T+u v). Although T_gua(u,v) and T_gua(v,u) are isomorphic to each other, only paths and nodes relevant for explanation are shown in this figure.