Figure 7
From: Efficient enumeration of monocyclic chemical graphs with given path frequencies
Illustration of defining the parent π ( G ) of a 1-augmented tree G . Three multi-trees T1=G−u w, T2=G−v z and T3=G−w z are obtained from G by removing a simple edge in the cycle. Each number on the left side of each vertex v in G indicates its signature σ∗(v) of {T w ∣w∈N(v)}. The code σ∗ for each pair of adjacent vertices in the cycle of G is given by σ∗(w,u)=((C,2,1),1,(C,2),1,(C,1),2,(C)), σ∗(u,w)=((C),2,(C,1),1,(C,2),1,(C,2,1)), σ∗(v,z)=((C,1),2,(C),1,(C,2,1),1,(C,2)), σ∗(z,v)=((C,2),1,(C,2,1),1,(C),2,(C,1)), σ∗(w,z)=((C,2,1),1,(C),2,(C,1),1,(C,2)), and σ∗(z,w)=((C,2),1,(C,1),2,(C),1,(C,2,1)). Then π(G) is defined to be T1=G−u w because σ∗(w,u) is maximum over all of these six codes.