diff --git a/Project.toml b/Project.toml
index 613aa19..17c5af9 100644
--- a/Project.toml
+++ b/Project.toml
@@ -26,7 +26,7 @@ KaHyPar = "0.3"
 StatsBase = "0.34"
 Suppressor = "0.2"
 LuxorGraphPlot = "0.5.1"
-TreeWidthSolver = "0.1.0"
+TreeWidthSolver = "0.2"
 julia = "1.9"
 
 [extras]
diff --git a/src/treewidth.jl b/src/treewidth.jl
index 74cfb7c..48659e0 100644
--- a/src/treewidth.jl
+++ b/src/treewidth.jl
@@ -55,9 +55,11 @@ ab, ab -> a
 """
 function exact_treewidth_method(incidence_list::IncidenceList{VT,ET}, log2_edge_sizes; α::TA = 0.0, temperature::TT = 0.0, nrepeat=1) where {VT,ET,TA,TT}
     indicies = collect(keys(incidence_list.e2v))
+    tensors = collect(keys(incidence_list.v2e))
     weights = [log2_edge_sizes[e] for e in indicies]
     line_graph = il2lg(incidence_list, indicies)
 
+    scalars = [i for i in tensors if isempty(incidence_list.v2e[i])]
     contraction_trees = Vector{Union{ContractionTree, VT}}()
 
     # avoid the case that the line graph is not connected
@@ -71,8 +73,9 @@ function exact_treewidth_method(incidence_list::IncidenceList{VT,ET}, log2_edge_
         contraction_tree = eo2ct(elimination_order, incidence_list, log2_edge_sizes, α, temperature, nrepeat)
         push!(contraction_trees, contraction_tree)
     end
-    
-    return reduce((x,y) -> ContractionTree(x, y), contraction_trees)
+
+    # add the scalars back to the contraction tree
+    return reduce((x,y) -> ContractionTree(x, y), contraction_trees ∪ scalars)
 end
 
 # transform incidence list to line graph
@@ -104,16 +107,16 @@ function eo2ct(elimination_order::EliminationOrder, incidence_list::IncidenceLis
     flag = contraction_tree_nodes[1]
 
     while !isempty(eo)
-        e = pop!(eo)
-        if haskey(incidence_list.e2v, e)
-            vs = incidence_list.e2v[e]
-            if length(vs) >= 2
-                sub_list = IncidenceList(Dict([v => incidence_list.v2e[v] for v in vs]); openedges=incidence_list.openedges)
-                sub_tree, scs, tcs = tree_greedy(sub_list, log2_edge_sizes; nrepeat=nrepeat, α=α, temperature=temperature)
-                vi = contract_tree!(incidence_list, sub_tree, log2_edge_sizes, scs, tcs)
-                contraction_tree_nodes[tensors_list[vi]] = st2ct(sub_tree, tensors_list, contraction_tree_nodes)
-                flag = vi
-            end
+        eliminated_vertices = pop!(eo) # e is a vector of vertices, which are eliminated at the same time
+        vs = unique!(vcat([incidence_list.e2v[ei] for ei in eliminated_vertices if haskey(incidence_list.e2v, ei)]...)) # the tensors to be contracted, since they are connected to the eliminated vertices
+        if length(vs) >= 2
+            sub_list_indices = unique!(vcat([incidence_list.v2e[v] for v in vs]...)) # the vertices connected to the tensors to be contracted
+            sub_list_open_indices = setdiff(sub_list_indices, eliminated_vertices) # the vertices connected to the tensors to be contracted but not eliminated
+            sub_list = IncidenceList(Dict([v => incidence_list.v2e[v] for v in vs]); openedges=sub_list_open_indices) # the subgraph of the contracted tensors
+            sub_tree, scs, tcs = tree_greedy(sub_list, log2_edge_sizes; nrepeat=nrepeat, α=α, temperature=temperature) # optmize the subgraph with greedy method
+            vi = contract_tree!(incidence_list, sub_tree, log2_edge_sizes, scs, tcs) # insert the contracted tensors back to the total graph
+            contraction_tree_nodes[tensors_list[vi]] = st2ct(sub_tree, tensors_list, contraction_tree_nodes)
+            flag = vi
         end
     end
 
diff --git a/test/treewidth.jl b/test/treewidth.jl
index 901e941..79b724e 100644
--- a/test/treewidth.jl
+++ b/test/treewidth.jl
@@ -34,4 +34,10 @@ using Test, Random
     cc = contraction_complexity(optcode, size_dict)
     @test cc.sc == 7
     @test decorate(eincode)(tensors...) ≈ decorate(optcode)(tensors...)
+
+    eincode = OMEinsumContractionOrders.EinCode([['a', 'b'], ['a', 'c', 'd'], ['b', 'c', 'e'], ['e'], ['f'], Char[]], ['a', 'f'])
+    tensors = tensors ∪ [fill(2.0,())]
+    optcode = optimize_exact_treewidth(optimizer, eincode, size_dict)
+    cc = contraction_complexity(optcode, size_dict)
+    @test decorate(eincode)(tensors...) ≈ decorate(optcode)(tensors...)
 end
\ No newline at end of file