diff --git a/src/coloring.jl b/src/coloring.jl
index 2746b30b..cf3d8f79 100644
--- a/src/coloring.jl
+++ b/src/coloring.jl
@@ -124,7 +124,7 @@ function star_coloring(
 
     for v in vertices_in_order
         for (w, index_vw) in neighbors_with_edge_indices(g, v)
-            !has_diagonal(g) || (v == w && continue)
+            augmented_graph(g) || (v == w && continue)
             color_w = color[w]
             iszero(color_w) && continue
             forbidden_colors[color_w] = v
@@ -139,7 +139,7 @@ function star_coloring(
             else
                 first_neighbor[color[w]] = (v, w, index_vw)
                 for (x, index_wx) in neighbors_with_edge_indices(g, w)
-                    !has_diagonal(g) || (w == x && continue)
+                    augmented_graph(g) || (w == x && continue)
                     color_x = color[x]
                     (x == v || iszero(color_x)) && continue
                     if x == hub[star[index_wx]]  # potential Case 2 (which is always false for trivial stars with two vertices, since the associated hub is negative)
@@ -184,7 +184,7 @@ function _treat!(
     color::AbstractVector{<:Integer},
 )
     for x in neighbors(g, w)
-        !has_diagonal(g) || (w == x && continue)
+        augmented_graph(g) || (w == x && continue)
         color_x = color[x]
         iszero(color_x) && continue
         forbidden_colors[color_x] = v
@@ -204,12 +204,12 @@ function _update_stars!(
     first_neighbor::AbstractVector{<:Tuple},
 )
     for (w, index_vw) in neighbors_with_edge_indices(g, v)
-        !has_diagonal(g) || (v == w && continue)
+        augmented_graph(g) || (v == w && continue)
         color_w = color[w]
         iszero(color_w) && continue
         x_exists = false
         for (x, index_wx) in neighbors_with_edge_indices(g, w)
-            !has_diagonal(g) || (w == x && continue)
+            augmented_graph(g) || (w == x && continue)
             if x != v && color[x] == color[v]  # vw, wx ∈ E
                 star_wx = star[index_wx]
                 hub[star_wx] = w  # this may already be true
@@ -286,16 +286,16 @@ function acyclic_coloring(
 
     for v in vertices_in_order
         for w in neighbors(g, v)
-            !has_diagonal(g) || (v == w && continue)
+            augmented_graph(g) || (v == w && continue)
             color_w = color[w]
             iszero(color_w) && continue
             forbidden_colors[color_w] = v
         end
         for w in neighbors(g, v)
-            !has_diagonal(g) || (v == w && continue)
+            augmented_graph(g) || (v == w && continue)
             iszero(color[w]) && continue
             for (x, index_wx) in neighbors_with_edge_indices(g, w)
-                !has_diagonal(g) || (w == x && continue)
+                augmented_graph(g) || (w == x && continue)
                 color_x = color[x]
                 iszero(color_x) && continue
                 if forbidden_colors[color_x] != v
@@ -320,16 +320,16 @@ function acyclic_coloring(
             end
         end
         for (w, index_vw) in neighbors_with_edge_indices(g, v)  # grow two-colored stars around the vertex v
-            !has_diagonal(g) || (v == w && continue)
+            augmented_graph(g) || (v == w && continue)
             color_w = color[w]
             iszero(color_w) && continue
             _grow_star!(v, w, index_vw, color_w, first_neighbor, forest)
         end
         for (w, index_vw) in neighbors_with_edge_indices(g, v)
-            !has_diagonal(g) || (v == w && continue)
+            augmented_graph(g) || (v == w && continue)
             iszero(color[w]) && continue
             for (x, index_wx) in neighbors_with_edge_indices(g, w)
-                !has_diagonal(g) || (w == x && continue)
+                augmented_graph(g) || (w == x && continue)
                 color_x = color[x]
                 (x == v || iszero(color_x)) && continue
                 if color_x == color[v]
diff --git a/src/decompression.jl b/src/decompression.jl
index a48c7e34..42bf51eb 100644
--- a/src/decompression.jl
+++ b/src/decompression.jl
@@ -547,7 +547,7 @@ function decompress!(
     end
 
     # Recover the diagonal coefficients of A
-    if has_diagonal(ag)
+    if !augmented_graph(ag)
         for i in axes(S, 1)
             if !iszero(S[i, i])
                 A[i, i] = B[i, color[i]]
@@ -616,7 +616,7 @@ function decompress!(
     end
 
     # Recover the diagonal coefficients of A
-    if has_diagonal(ag)
+    if !augmented_graph(ag)
         if uplo == :L
             for i in diagonal_indices
                 # A[i, i] is the first element in column i
diff --git a/src/graph.jl b/src/graph.jl
index 9b53a36d..84c3353f 100644
--- a/src/graph.jl
+++ b/src/graph.jl
@@ -202,7 +202,7 @@ end
 ## Adjacency graph
 
 """
-    AdjacencyGraph{T,has_diagonal}
+    AdjacencyGraph{T,augmented_graph}
 
 Undirected graph without self-loops representing the nonzeros of a symmetric matrix (typically a Hessian matrix).
 
@@ -213,18 +213,18 @@ The adjacency graph of a symmetric matrix `A ∈ ℝ^{n × n}` is `G(A) = (V, E)
 
 # Constructors
 
-    AdjacencyGraph(A::SparseMatrixCSC; has_diagonal::Bool=true)
+    AdjacencyGraph(A::SparseMatrixCSC; augmented_graph::Bool=false)
 
 # Fields
 
-- `S::SparsityPatternCSC{T}`: Underlying sparsity pattern, whose diagonal is empty whenever `has_diagonal` is `false`
+- `S::SparsityPatternCSC{T}`: Underlying sparsity pattern, which represents an augmented graph whenever `augmented_graph` is `true`. Here, "augmented graph" means the sparsity pattern of the augmented matrix `H = [0 Jᵀ; J 0]`.
 - `edge_to_index::Vector{T}`: A vector mapping each nonzero of `S` to a unique edge index (ignoring diagonal and accounting for symmetry, so that `(i, j)` and `(j, i)` get the same index)
 
 # References
 
 > [_What Color Is Your Jacobian? SparsityPatternCSC Coloring for Computing Derivatives_](https://epubs.siam.org/doi/10.1137/S0036144504444711), Gebremedhin et al. (2005)
 """
-struct AdjacencyGraph{T<:Integer,has_diagonal}
+struct AdjacencyGraph{T<:Integer,augmented_graph}
     S::SparsityPatternCSC{T}
     edge_to_index::Vector{T}
 end
@@ -234,24 +234,24 @@ Base.eltype(::AdjacencyGraph{T}) where {T} = T
 function AdjacencyGraph(
     S::SparsityPatternCSC{T},
     edge_to_index::Vector{T}=build_edge_to_index(S);
-    has_diagonal::Bool=true,
+    augmented_graph::Bool=false,
 ) where {T}
-    return AdjacencyGraph{T,has_diagonal}(S, edge_to_index)
+    return AdjacencyGraph{T,augmented_graph}(S, edge_to_index)
 end
 
-function AdjacencyGraph(A::SparseMatrixCSC; has_diagonal::Bool=true)
-    return AdjacencyGraph(SparsityPatternCSC(A); has_diagonal)
+function AdjacencyGraph(A::SparseMatrixCSC; augmented_graph::Bool=false)
+    return AdjacencyGraph(SparsityPatternCSC(A); augmented_graph)
 end
 
-function AdjacencyGraph(A::AbstractMatrix; has_diagonal::Bool=true)
-    return AdjacencyGraph(SparseMatrixCSC(A); has_diagonal)
+function AdjacencyGraph(A::AbstractMatrix; augmented_graph::Bool=false)
+    return AdjacencyGraph(SparseMatrixCSC(A); augmented_graph)
 end
 
 pattern(g::AdjacencyGraph) = g.S
 edge_indices(g::AdjacencyGraph) = g.edge_to_index
 nb_vertices(g::AdjacencyGraph) = pattern(g).n
 vertices(g::AdjacencyGraph) = 1:nb_vertices(g)
-has_diagonal(::AdjacencyGraph{T,hd}) where {T,hd} = hd
+augmented_graph(::AdjacencyGraph{T,ag}) where {T,ag} = ag
 
 function neighbors(g::AdjacencyGraph, v::Integer)
     S = pattern(g)
@@ -267,17 +267,17 @@ function neighbors_with_edge_indices(g::AdjacencyGraph, v::Integer)
     return zip(neighbors_v, edges_indices_v)
 end
 
-degree(g::AdjacencyGraph{T,false}, v::Integer) where {T} = g.S.colptr[v + 1] - g.S.colptr[v]
+degree(g::AdjacencyGraph{T,true}, v::Integer) where {T} = g.S.colptr[v + 1] - g.S.colptr[v]
 
-function degree(g::AdjacencyGraph{T,true}, v::Integer) where {T}
+function degree(g::AdjacencyGraph{T,false}, v::Integer) where {T}
     neigh = neighbors(g, v)
     has_selfloop = insorted(v, neigh)
     return g.S.colptr[v + 1] - g.S.colptr[v] - has_selfloop
 end
 
-nb_edges(g::AdjacencyGraph{T,false}) where {T} = nnz(g.S) ÷ 2
+nb_edges(g::AdjacencyGraph{T,true}) where {T} = nnz(g.S) ÷ 2
 
-function nb_edges(g::AdjacencyGraph{T,true}) where {T}
+function nb_edges(g::AdjacencyGraph{T,false}) where {T}
     ne = 0
     for v in vertices(g)
         ne += degree(g, v)
@@ -290,7 +290,7 @@ minimum_degree(g::AdjacencyGraph) = minimum(Base.Fix1(degree, g), vertices(g))
 
 function has_neighbor(g::AdjacencyGraph, v::Integer, u::Integer)
     for w in neighbors(g, v)
-        !has_diagonal(g) || (v == w && continue)
+        augmented_graph(g) || (v == w && continue)
         if w == u
             return true
         end
diff --git a/src/interface.jl b/src/interface.jl
index b8d4ce54..cc57bcc7 100644
--- a/src/interface.jl
+++ b/src/interface.jl
@@ -279,7 +279,7 @@ function _coloring(
     symmetric_pattern::Bool;
     forced_colors::Union{AbstractVector{<:Integer},Nothing}=nothing,
 )
-    ag = AdjacencyGraph(A; has_diagonal=true)
+    ag = AdjacencyGraph(A; augmented_graph=false)
     color_and_star_set_by_order = map(algo.orders) do order
         vertices_in_order = vertices(ag, order)
         return star_coloring(ag, vertices_in_order, algo.postprocessing; forced_colors)
@@ -300,7 +300,7 @@ function _coloring(
     decompression_eltype::Type{R},
     symmetric_pattern::Bool,
 ) where {R}
-    ag = AdjacencyGraph(A; has_diagonal=true)
+    ag = AdjacencyGraph(A; augmented_graph=false)
     color_and_tree_set_by_order = map(algo.orders) do order
         vertices_in_order = vertices(ag, order)
         return acyclic_coloring(ag, vertices_in_order, algo.postprocessing)
@@ -323,7 +323,7 @@ function _coloring(
     forced_colors::Union{AbstractVector{<:Integer},Nothing}=nothing,
 ) where {R}
     A_and_Aᵀ, edge_to_index = bidirectional_pattern(A; symmetric_pattern)
-    ag = AdjacencyGraph(A_and_Aᵀ, edge_to_index; has_diagonal=false)
+    ag = AdjacencyGraph(A_and_Aᵀ, edge_to_index; augmented_graph=true)
     outputs_by_order = map(algo.orders) do order
         vertices_in_order = vertices(ag, order)
         _color, _star_set = star_coloring(
@@ -370,7 +370,7 @@ function _coloring(
     symmetric_pattern::Bool,
 ) where {R}
     A_and_Aᵀ, edge_to_index = bidirectional_pattern(A; symmetric_pattern)
-    ag = AdjacencyGraph(A_and_Aᵀ, edge_to_index; has_diagonal=false)
+    ag = AdjacencyGraph(A_and_Aᵀ, edge_to_index; augmented_graph=true)
     outputs_by_order = map(algo.orders) do order
         vertices_in_order = vertices(ag, order)
         _color, _tree_set = acyclic_coloring(ag, vertices_in_order, algo.postprocessing)
diff --git a/src/order.jl b/src/order.jl
index aab5bb23..20884c6a 100644
--- a/src/order.jl
+++ b/src/order.jl
@@ -334,7 +334,7 @@ function vertices(
 
         π[index] = u
         for v in neighbors(g, u)
-            !has_diagonal(g) || (u == v && continue)
+            augmented_graph(g) || (u == v && continue)
             dv = degrees[v]
             dv == -1 && continue
             update_bucket!(db, v, dv; degtype, direction)
diff --git a/src/postprocessing.jl b/src/postprocessing.jl
index 217b663b..8174d863 100644
--- a/src/postprocessing.jl
+++ b/src/postprocessing.jl
@@ -13,7 +13,7 @@ function postprocess!(
     color_used = zeros(Bool, nb_colors)
 
     # nonzero diagonal coefficients force the use of their respective color (there can be no neutral colors if the diagonal is fully nonzero)
-    if has_diagonal(g)
+    if !augmented_graph(g)
         for i in axes(S, 1)
             if !iszero(S[i, i])
                 color_used[color[i]] = true
diff --git a/src/result.jl b/src/result.jl
index 7a3e3b68..89da22d6 100644
--- a/src/result.jl
+++ b/src/result.jl
@@ -411,7 +411,7 @@ function TreeSetColoringResult(
     diagonal_nzind = T[]
     ndiag = 0
 
-    if has_diagonal(ag)
+    if !augmented_graph(ag)
         for j in axes(S, 2)
             for k in nzrange(S, j)
                 i = rv[k]
diff --git a/test/graph.jl b/test/graph.jl
index 9c8f66c1..e9784e56 100644
--- a/test/graph.jl
+++ b/test/graph.jl
@@ -166,7 +166,7 @@ end;
     @test degree(g, 7) == 6
     @test degree(g, 8) == 6
 
-    g = AdjacencyGraph(transpose(A) * A; has_diagonal=false)
+    g = AdjacencyGraph(transpose(A) * A; augmented_graph=true)
     # wrong degree
     @test degree(g, 1) == 4
     @test degree(g, 2) == 4
diff --git a/test/order.jl b/test/order.jl
index 838c32a9..9cae16f1 100644
--- a/test/order.jl
+++ b/test/order.jl
@@ -61,16 +61,14 @@ end;
 end;
 
 @testset "LargestFirst" begin
-    for has_diagonal in (false, true)
-        A = sparse([
-            0 1 0 0
-            1 0 1 1
-            0 1 0 1
-            0 1 1 0
-        ])
-        ag = AdjacencyGraph(A; has_diagonal)
-        @test vertices(ag, LargestFirst()) == [2, 3, 4, 1]
-    end
+    A = sparse([
+        0 1 0 0
+        1 0 1 1
+        0 1 0 1
+        0 1 1 0
+    ])
+    ag = AdjacencyGraph(A)
+    @test vertices(ag, LargestFirst()) == [2, 3, 4, 1]
 
     A = sparse([
         1 1 0 0