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

$$\mathcal {G}_{s}$$ 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 $$\left \{ \ell _{2}^{1},\ell _{1}^{2} \right \}$$
R7(v6)=(0,0,0)
$$R_{7}(v_{8}) = (\mathcal {C}_{8}, 0, 0)$$
R7(v9)=(0,0,0)
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