Annihilation = {
    #[x_] -> #[y_],
    #[y_] -> #[x_],
    f1n_[OrderlessPatternSequence[a_,  #[x_]]],
    f2n_[OrderlessPatternSequence[b_,  #[x_]]],
    f3n_[OrderlessPatternSequence[c_,  #[y_]]],
    f4n_[OrderlessPatternSequence[d_,  #[y_]]]
    }:> Splice[
        Join[
            If[
                f1n===f4n===UndirectedEdge,
                {a\[UndirectedEdge]d,d\[UndirectedEdge]a},
                {f1n[a,d],f4n[d,a]}
            ],
            If[
                f2n===f3n===UndirectedEdge,
                {b\[UndirectedEdge]c,c\[UndirectedEdge]b},
                {f2n[b,c],f3n[c,b]}
            ]
        ]
    ]&
δAnnihilation={
    δ[x_] -> δ[y_],
    δ[y_] -> δ[x_],
    f1n_[OrderlessPatternSequence[a_,  δ[x_]]],
    f2n_[OrderlessPatternSequence[b_,  δ[x_]]],
    f3n_[OrderlessPatternSequence[c_,  δ[y_]]],
    f4n_[OrderlessPatternSequence[d_,  δ[y_]]]
    }:> Splice[
        Join[
            If[
                False&&f1n===f3n===UndirectedEdge,
                {a\[UndirectedEdge]c},
                {f1n[a,c],f3n[c,a]}
            ],
            If[
                False&&f2n===f4n===UndirectedEdge,
                {b\[UndirectedEdge]d},
                {f2n[b,d],f4n[d,b]}
            ]
        ]
    ]
γAnnihilation= {
    γ[x_] -> γ[y_],
    γ[y_] -> γ[x_],
    f1n_[OrderlessPatternSequence[a_,  γ[x_]]],
    f2n_[OrderlessPatternSequence[b_,  γ[x_]]],
    f3n_[OrderlessPatternSequence[c_,  γ[y_]]],
    f4n_[OrderlessPatternSequence[d_,  γ[y_]]]
    }:> Splice[
        Join[
            If[
                f1n===f4n===UndirectedEdge,
                {a\[UndirectedEdge]d},
                {f1n[a,d],f4n[d,a]}
            ],
            If[
                f2n===f3n===UndirectedEdge,
                {b\[UndirectedEdge]c},
                {f2n[b,c],f3n[c,b]}
            ]
        ]
    ]
Duplication = {
γ[x_]->δ[y_],
δ[y_]->γ[x_],
f1n_[OrderlessPatternSequence[a_,γ[x_]]],
f2n_[OrderlessPatternSequence[b_,γ[x_]]],
f3n_[OrderlessPatternSequence[c_,δ[y_]]],
f4n_[OrderlessPatternSequence[d_,δ[y_]]]
}:> Module[{γ1,γ2,δ1,δ2},Splice[
Join[
{
γ[γ1]\[UndirectedEdge]δ[δ1],
γ[γ1]\[UndirectedEdge]δ[δ2],
γ[γ2]\[UndirectedEdge]δ[δ1],
γ[γ2]\[UndirectedEdge]δ[δ2]
},
{γ[γ1]->c,f3n[c,γ[γ1]], γ[γ2]->d,f4n[d,γ[γ2]]},
If[a===b,{δ[δ1]->δ[δ2],δ[δ2]->δ[δ1]}
,{δ[δ1]->a,f1n[a,δ[δ1]], δ[δ2]->b,f2n[b,δ[δ2]]}]
]
]]
RemoveCombinators={δ->Identity, γ->Identity}
Labels=SubsetReplace[VertexList@#,{{γ[x_]}->x->Placed["γ",Center],{δ[x_]}->x->Placed["δ",Center],{_}->Nothing}]&

A=Module[{a},{γ[a]->x, γ[a]\[UndirectedEdge]γ[a], γ[a]\[UndirectedEdge]γ[a]}]
B=Module[{y,z,w},{γ[y]->x,δ[z]->γ[y],γ[w]->δ[z],δ[z]\[UndirectedEdge]γ[w],γ[w]\[UndirectedEdge]γ[y]}]
IDK=Flatten[SubsetReplace[#,{δAnnihilation, γAnnihilation,Duplication}]]&
R=Graph[#/.RemoveCombinators,VertexLabels->Labels[#],VertexSize->0.25]&
R[A]
R[B]

G=Module[{b,a,y,z,w},{x\[UndirectedEdge]γ[b],γ[a]\[UndirectedEdge]γ[a],γ[a]\[UndirectedEdge]γ[a],γ[a]->γ[b],γ[b]->γ[y],γ[y]->γ[b],δ[z]->γ[y],γ[w]->δ[z],δ[z]\[UndirectedEdge]γ[w],γ[w]\[UndirectedEdge]γ[y]}]

G=Module[{z}, EdgeAdd[GraphUnion[A/.x\[UndirectedEdge]y_->γ[z]\[UndirectedEdge]y, B/.x\[UndirectedEdge]y_->γ[z]\[UndirectedEdge]y],{x\[UndirectedEdge]γ[z]}]]

R[G]
G2=SubsetReplace[G,{γAnnihilation}]
R[G2]
G3=SubsetReplace[G2,{Duplication}]
R[G3]
G4=SubsetReplace[G3,{δAnnihilation}]
R[G4]
G5=SubsetReplace[G4,{γAnnihilation}]
R[G5]