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I have used the functions EdgeDelete and EdgeAdd to expand the graph. The code generates new vertex names by incrementing the largest vertex name in the graph.

nextVertexNames[g_] := Max[VertexList[g]] + {{1, 2, 3}, {4, 5, 6}}
replaceTripod[g_, v_] := Module[{
   oldNeighbors = DeleteCases[VertexComponent[g, v, 1], v], 
   newNeighbors = nextVertexNames[g], go = g
   },
  EdgeAdd[
   EdgeDelete[go, v <-> _],
   Flatten@{
     UndirectedEdge[v, #] & /@ First[newNeighbors],
     UndirectedEdge @@@ Partition[Riffle @@ newNeighbors, {2}, 1, 1],
     UndirectedEdge @@@ Thread[{Last@newNeighbors, oldNeighbors}]}
   ]
  ]
replaceTripods[g_] := 
  Fold[replaceTripod, g, 
   Extract[VertexList[g], Position[VertexDegree[g], 3]]];

For your case, use replaceTripods[g] to replace all tripods in the graph g.