# Vertex Labeling and Routing for Farey-Type Symmetrically-Structured Graphs
## 5 Routing of \(GF(t,k)\) and \(EF(t,k)\)  
Although the labels of the three initial vertices in GFG and EFG are slightly different from the labels of the two initial vertices in FG, the shortest paths protocol in GFG/EFG also benefit from the routing algorithm in FG. From the generation mechanism, GFG/EFG is divided into three groups by symmetry, and each group is consisted of \(k\) or \(t\times k\) petals, and furthermore vertices in each group can also be divided into three sets by the differences in distance, similar to the equivalent division for FG: sets \(V^{x}(t)\), \(V^{xy}(t)\) and \(V^{y}(t)\). **Property 11**. : _(The characteristic of shortest paths when two vertices are located in different groups)_  
_If \(a_{1}\
eq a_{2}\), the initial vertex \(\{0,1,2\}/\{a_{1},a_{2}\}\) is on the shortest paths between \(a_{1}.b_{1}.c_{1}.d_{1}\) and \(a_{2}.b_{2}.c_{2}.d_{2}\) of GFG/EFG, if_  
_(a) \(a_{1}.b_{1}.c_{1}\in V_{1}^{x}(t)\cup V_{1}^{xy}(t)\) and \(a_{2}.b_{2}.c_{2}\in V_{2}^{x}(t)\cup V_{2}^{xy}(t)\),_  
_(b) or \(a_{1}.b_{1}.c_{1}\in V_{2}^{x}(t)\cup V_{2}^{xy}(t)\) and \(a_{2}.b_{2}.c_{2}\in V_{1}^{x}(t)\cup V_{1}^{xy}(t)\),_  
_(c) or \(a_{1}.b_{1}.c_{1}\in V_{1}^{x}(t)\) and \(a_{2}.b_{2}.c_{2}\in V_{2}^{y}(t)/\{a_{2}\}\),_  
_(d) or \(a_{1}.b_{1}.c_{1}\in V_{2}^{x}(t)\) and \(a_{2}.b_{2}.c_{2}\in V_{1}^{y}(t)/\{a_{1}\}\)._  
_The two initial vertices \(a_{1}\) and \(a_{2}\) are on the shortest paths, if_  
_(a) \(a_{1}.b_{1}.c_{1}\in V_{1}^{y}(t)\) and \(a_{2}.b_{2}.c_{2}\in V_{2}^{y}(t)\),_  
_(b) or \(a_{1}.b_{1}.c_{1}\in V_{2}^{y}(t)\) and \(a_{2}.b_{2}.c_{2}\in V_{1}^{y}(t)\)._  
_The shortest paths pass \(\{0,1,2\}/\{a_{1},a_{2}\}\), \(a_{1}\) and \(a_{2}\) simultaneously, if_  
_(a) \(a_{1}.b_{1}.c_{1}\in V_{1}^{y}(t)\) and \(a_{2}.b_{2}.c_{2}\in V_{2}^{xy}(t)\),_  
_(b) or \(a_{1}.b_{1}.c_{1}\in V_{2}^{y}(t)\) and \(a_{2}.b_{2}.c_{2}\in V_{1}^{xy}(t)\)._  
Figure 7: The projected graph of \(GF(2,2)/EF(2,2)\). **Proof. ** From Property 9, \(a_{1}.b_{1}.c_{1}.d_{1}\) and \(a_{2}.b_{2}.c_{2}.d_{2}\) are projected as \(a_{1}.b_{1}.c_{1}\) and \(a_{2}.b_{2}.c_{2}\), which are located in different subgraphs \(F_{1}(t)\) and \(F_{2}(t)\) of Farey graphs \(F(t+1)\). Following from the proof of Property 3, the conclusions can easily be deduced. \(\square\)  
**Property 12. **_(The characteristic of shortest paths when two vertices lie in different slices of same group) If \(a.b_{1}.c_{1}.d_{1}\) and \(a.b_{2}.c_{2}.d_{2}\) are located in different petals, the shortest paths between them are positioned in two slices which are connected by two common vertices._  
**Proof. ** From the generating algorithm, if \(a.b_{1}.c_{1}.d_{1}\) and \(a.b_{2}.c_{2}.d_{2}\) are located in different petals or subpetals, then the neighbor sets of \(a.b_{1}.c_{1}.d_{1}\) and \(a.b_{2}.c_{2}.d_{2}\) are ascertained by Property 8. When two common neighbors are created during the same step, then \(a.b_{1}.c_{1}.d_{1}\) and \(a.b_{2}.c_{2}.d_{2}\) are positioned in different slices, and the two slices can be determined by Property 10. Supposing that the first two common neighbors are \(a.b_{3}.c_{3}.d_{3}\) and \(a. (b_{3}+1).c_{4}.d_{4}\), the two slices are rooted in them and belong to \(F(b_{1}-b_{3})\) and \(F(b_{2}-b_{3})\). Compared with the construction schematic diagram in Figure 2, the linking of \(F(b_{1}-b_{3})\) and \(F(b_{2}-b_{3})\) is slightly different. Name the initial vertices of \(F(b_{1}-b_{3})\) and \(F(b_{2}-b_{3})\) as \(X_{1}\), \(X_{2}\) and \(Y_{1}\), \(Y_{2}\), respectively. \(F(b_{1}-b_{3})\) and \(F(b_{2}-b_{3})\) are linked exactly by merging \(X_{1}\) and \(Y_{1}\) into \(a.b_{3}.c_{3}.d_{3}\) and linking \(X_{2}\) and \(Y_{2}\) into \(a.(b_{3}+1).c_{4}.d_{4}\).