Stable Set Bonding in Perfect Graphs and Parity Graphs

Let G1 = (V1 ∪ S1, E1) and G2 = (V2 ∪ S2, E2) be connected graphs which each have stable sets S1 (resp. S2) of the same size. Let ΦS be the operation which forms G = (V, E) from G1 and G2 by identification of S1 and S2, where S ⊆ V corresponds to S1 and S2. If all minimal chains in G1 and G2, linking v to w for v, w ∈ S have the same parity, and if H1 and H2 are parity graphs where G1 ΦSH2, H1 ΦSG2, and H1 ΦSH2 are perfect graphs then G1 ΦSG2, is also perfect. This leads to a new composition operation which preserves perfection.