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Figure 10 | Journal of Cheminformatics

Figure 10

From: Efficient enumeration of monocyclic chemical graphs with given path frequencies

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 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 û of u (i.e., dfs(û)=dfs(u)+| T gua ( u , v ) |). Similarly Tgua(u,v) contains a copy v ̂ of v (i.e., dfs( v ̂ )=dfs(v)| T gua ( u , v ) |). Condition (3)-(ii) requires dfs(û)dfs(v) (otherwise we can prove that ( v ̂ ,û) is a lexicographically smaller pair such that T+ v ̂ û 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.

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