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Table 4 Three-tuples used to determine the edges and labels for the example in Fig. 14

From: Backtracking-based dynamic programming for resolving transmit ambiguities in WSN localization

\(\mathcal {G}_{s}\) Constraint cliques Three-tuples of the vertices of \(\mathcal {G}_{i}\) Action/comment
\(\mathcal {G}_{1}\) \(\mathcal {C}_{1}\) \(R_{1}(v_{1}) = (\mathcal {C}_{7}, \mathcal {C}_{5}, 0)\) R1(v1)≠R1(v8) (v1)≠(v8)
   \(R_{1}(v_{8}) = (\mathcal {C}_{8}, \mathcal {C}_{9}, 0)\)  
\(\mathcal {G}_{2}\) \(\mathcal {C}_{2}\) \(R_{2}(v_{2}) = (\mathcal {C}_{7}, \mathcal {C}_{5}, 0)\) R2(v2)≠R2(v3) (v2)≠(v3)
   \(R_{2}(v_{3}) = (\mathcal {C}_{8}, \mathcal {C}_{9}, 0)\)  
\(\mathcal {G}_{3}\) \(\mathcal {C}_{3}\) \(R_{3}(v_{5}) = (\mathcal {C}_{8}, \mathcal {C}_{6}, 0)\) R3(v5)≠R3(v9) (v5)≠(v9)
   \(R_{3}(v_{9}) = (\mathcal {C}_{7}, \mathcal {C}_{5}, 0)\)  
\(\mathcal {G}_{4}\) \(\mathcal {C}_{4}\) \(R_{4}(v_{4}) = (\mathcal {C}_{8}, \mathcal {C}_{6}, 0)\) R4(v4)≠R4(v6) (v4)≠(v6)
   \(R_{4}(v_{6}) = (\mathcal {C}_{7}, \mathcal {C}_{9}, 0)\)  
\(\mathcal {G}_{5}\) \(\mathcal {C}_{1},\) \(\mathcal {C}_{2}\) \(R_{5}(v_{1}) = (\mathcal {C}_{7}, \mathcal {C}_{5}, 0)\) R5(v i )R5(v j ),i,j{1,2,3,8} complete graph
   \(R_{5}(v_{2}) = (\mathcal {C}_{7}, \mathcal {C}_{5}, 0)\) R5(v1)=R5(v2) (v1)=(v2)
   \(R_{5}(v_{3}) = (\mathcal {C}_{8}, \mathcal {C}_{9}, 0)\) R5(v3)=R5(v8) (v3)=(v8)
   \(R_{5}(v_{8}) = (\mathcal {C}_{8}, \mathcal {C}_{9}, 0)\)  
\(\mathcal {G}_{6}\) \(\mathcal {C}_{3}\), \(\mathcal {C}_{4}\), \(R_{6}(v_{4}) = (\mathcal {C}_{8}, 0, 0)\) R6(v4)R6(v5)(v4,v5)
   \(R_{6}(v_{5}) = (\mathcal {C}_{8}, 0, 0)\) R6(v6)R6(v9)(v6,v9)
   \(R_{6}(v_{6}) = (\mathcal {C}_{7}, \mathcal {C}_{9}, 0)\) R6(v4)=R6(v5) (v4)=(v5)
   \(R_{6}(v_{9}) = (\mathcal {C}_{7}, \mathcal {C}_{5}, 0)\)  
\(\mathcal {G}_{7}\) \(\mathcal {C}_{1}\), \(\mathcal {C}_{2}\), \(\mathcal {C}_{3}\), \(\mathcal {C}_{4}\), , \(\mathcal {C}_{6}\),, R7(v1)=(0,0,0) R7(v1)=R7(v2)=R7(v6)=R7(v9) (v1)=(v2)=(v6)=(v9)
   R7(v2)=(0,0,0) R7(v3)=R7(v4)=R7(v5)=R7(v8) (v3)=(v4)=(v5)=(v8)
   \(R_{7}(v_{3}) = (\mathcal {C}_{8}, 0, 0)\) R7(v i )R7(v j ),(i,j){1,2,6,9}2,i>j (v i ,v j )(i,j){1,2,6,9}2,i>j
   \(R_{7}(v_{4}) = (\mathcal {C}_{8}, 0, 0)\) R7(v i )R7(v j ),(i,j){3,4,5,8}2,i>j (v i ,v j )(i,j){3,4,5,8}2,i>j
   \(R_{7}(v_{5}) = (\mathcal {C}_{8}, 0, 0)\) ∙ Note that the last two operations introduce edges between the labels\(\left \{ \ell _{1}^{1}, \ell _{2}^{2}, \ell _{2}^{3} \right \}\) and between the labels \(\left \{ \ell _{2}^{1},\ell _{1}^{2} \right \}\)
   \(R_{7}(v_{8}) = (\mathcal {C}_{8}, 0, 0)\)  
  1. The edge sets of the leaves are given directly, as the leaves are chosen to be cliques
  2. The red-circled cliques are binding SC cliques, i.e., fully included SC cliques that have not been included before by either of the child nodes