g1=-CanonicalForm[BSeedGraph[3,{2,2}]]
g2=CanonicalForm[BSeedGraph[5,{3,3}]]

Fuse[g1_Graph,g2_Graph,c_] := Module[
  {mg,l1,l2,l,i,pos1,pos2,i1,i2,j1,j2},
  l1=Length[g1];
  l2=Length[g2];
  l=l1+l2;
  mg=g1~Join~(g2 /. i_Integer :> l1+i)~Join~Graph[OV,c[l+1]];
  pos1=First /@ Position[g1,c];
  pos2=l1+First /@ Position[g2,c];
  Sum[
    k1=First[mg[[j1=pos1[[i1]] ]] ];
    k2=First[mg[[j2=pos2[[i2]] ]] ];
    Fold[ReplacePart[#1,Last[#2],First[#2]]&, mg, {
      {j1,  Dashed[k1,l+1]},
      {j2,  Dashed[k2,l+1]},
      {l+1, OV[j1,j2,l+2]}
    }],
    {i1,Length[pos1]},{i2,Length[pos2]}
  ]
]
