The list of "new edge" graphs are formed as follows: Let ab and cd be disjoint edges of G. Then the associated "new edge" graph H is G with the edges ab and cd removed, the vertices e and f added, and the new edges ae, be, cf, df, and ef added.
i1 : newEdges graph {{a,b},{c,d}}
o1 = {Graph{0 => set {4} }}
1 => set {4}
2 => set {5}
3 => set {5}
4 => set {0, 1, 5}
5 => set {2, 3, 4}
o1 : List
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If the List input format is used, then one should use care as the list may contain isomorphic pairs.