diff --git a/src/bruteforce.jl b/src/bruteforce.jl
index e62ba11b..adcf597c 100644
--- a/src/bruteforce.jl
+++ b/src/bruteforce.jl
@@ -5,12 +5,12 @@ A brute force method to find the best configuration of a problem.
 """
 struct BruteForce end
 
-function findbest(problem::AbstractProblem, ::BruteForce)
+function findbest(problem::AbstractProblem, ::BruteForce; atol=eps(Float64), rtol=eps(Float64))
     best_size = Inf
     best_configs = Vector{Int}[]
     configs = Iterators.product([flavors(problem) for i in 1:num_variables(problem)]...)
     for (size, config) in energy_multi(problem, configs)
-        if size == best_size
+        if isapprox(size, best_size; atol, rtol)
             push!(best_configs, collect(config))
         elseif size < best_size[1]
             best_size = Float64(size)
@@ -19,4 +19,4 @@ function findbest(problem::AbstractProblem, ::BruteForce)
         end
     end
     return best_configs
-end
\ No newline at end of file
+end
diff --git a/src/topology.jl b/src/topology.jl
index 610880ba..b744cc95 100644
--- a/src/topology.jl
+++ b/src/topology.jl
@@ -42,13 +42,12 @@ $TYPEDEF
 A unit disk graph is a graph in which the vertices are points in a plane and two vertices are connected by an edge if and only if the Euclidean distance between them is at most a given radius.
 
 ### Fields
-- `n::Int`: the number of vertices
 - `locations::Vector{NTuple{D, T}}`: the locations of the vertices
-- `radius::T`: the radius of the unit disk
+- `radius::Float64`: the radius of the unit disk
 """
 struct UnitDiskGraph{D, T} <: Graphs.AbstractGraph{Int}
     locations::Vector{NTuple{D, T}}
-    radius::T
+    radius::Float64
 end
 Base.:(==)(a::UnitDiskGraph, b::UnitDiskGraph) = a.locations == b.locations && a.radius == b.radius
 Graphs.nv(g::UnitDiskGraph) = length(g.locations)
@@ -71,9 +70,7 @@ end
 function all_edges(g::UnitDiskGraph)
     edges = Graphs.SimpleEdge{Int}[]
     for i in 1:nv(g), j in i+1:nv(g)
-        if sum(abs2, g.locations[i] .- g.locations[j]) ≤ g.radius^2
-            push!(edges, Graphs.SimpleEdge(i, j))
-        end
+        has_edge(g, i, j) && push!(edges, Graphs.SimpleEdge(i, j))
     end
     return edges
 end
@@ -95,7 +92,7 @@ Base.eltype(::Type{Graphs.SimpleGraphs.SimpleEdgeIter{UnitDiskGraph{D,T}}}) wher
         end
 
         # return the next edge if it exists
-        if sum(abs2, g.locations[i] .- g.locations[j]) ≤ g.radius^2
+        if has_edge(g, i, j)
             e = Graphs.SimpleEdge(i, j)
             state = (i, j + 1)
             return e, state
@@ -107,31 +104,26 @@ Base.eltype(::Type{Graphs.SimpleGraphs.SimpleEdgeIter{UnitDiskGraph{D,T}}}) wher
     return nothing
 end
 
-"""
-$TYPEDEF
+function Graphs.induced_subgraph(g::UnitDiskGraph, vlist::AbstractVector{<:Integer})
+    return UnitDiskGraph(g.locations[vlist], g.radius), vlist
+end
 
-A grid graph is a graph in which the vertices are arranged in a grid and two vertices are connected by an edge if and only if they are adjacent in the grid.
+function Graphs.neighbors(g::UnitDiskGraph, i::Int)
+    [j for j in 1:nv(g) if i != j && has_edge(g, i, j)]
+end
 
-### Fields
-- `grid::BitMatrix`: a matrix of booleans, where `true` indicates the presence of an edge.
-- `radius::Float64`: the radius of the unit disk
 """
-struct GridGraph <: Graphs.AbstractGraph{Int}
-    grid::BitMatrix
-    radius::Float64
+GridGraph is a unit disk graph with integer coordinates.
+"""
+const GridGraph{D} = UnitDiskGraph{D, Int}
+function GridGraph(matrix::AbstractMatrix{Bool}, radius)
+    return UnitDiskGraph(vec(getfield.(findall(matrix), :I)), Float64(radius))
 end
-Base.:(==)(a::GridGraph, b::GridGraph) = a.grid == b.grid && a.radius == b.radius
-Graphs.nv(g::GridGraph) = sum(g.grid)
-Graphs.vertices(g::GridGraph) = 1:nv(g)
-Graphs.ne(g::GridGraph) = length(Graphs.edges(g))
-function Graphs.edges(g::GridGraph)
-    udg = UnitDiskGraph([Float64.(x.I) for x in findall(g.grid)], g.radius)
-    return Graphs.edges(udg)
+function GridGraph(locations::AbstractVector{NTuple{D, Int}}, radius::Float64) where {D}
+    return UnitDiskGraph(locations, radius)
 end
 
-# not implemented
-# struct PlanarGraph <: Graphs.AbstractGraph{Int}
-# end
+# TODO: implement the planar graph
 
 ##### Extra interfaces #####
 _vec(e::Graphs.SimpleEdge) = [src(e), dst(e)]
@@ -161,4 +153,4 @@ function _add_edge_weight!(g::HyperGraph, c::HyperEdge{Int}, J, weight)
     end
     push!(g.edges, c)
     push!(J, weight)
-end
\ No newline at end of file
+end
diff --git a/test/rules/rules.jl b/test/rules/rules.jl
index 3c4a5d1a..d56d119f 100644
--- a/test/rules/rules.jl
+++ b/test/rules/rules.jl
@@ -84,6 +84,7 @@ end
         # reduce and solve
         result = reduceto(target_type, source)
         target = target_problem(result)
+        @test target isa target_type
         best_target = findbest(target, BruteForce())
 
         # extract the solution
diff --git a/test/rules/spinglass_sat.jl b/test/rules/spinglass_sat.jl
index 0f792dc5..3f6c068a 100644
--- a/test/rules/spinglass_sat.jl
+++ b/test/rules/spinglass_sat.jl
@@ -82,8 +82,8 @@ end
         d = x ∨ (c ∧ ¬z)
     end
     sg, vars = ProblemReductions.circuit2spinglass(circuit)
-    indexof(x) = findfirst(==(x), vars)
-    gadget = LogicGadget(sg, indexof.([:x, :y, :z]), [indexof(:d)])
+    indexof1(x) = findfirst(==(x), vars)
+    gadget = LogicGadget(sg, indexof1.([:x, :y, :z]), [indexof1(:d)])
     tb = truth_table(gadget; variables=vars)
     @test tb.values == vec([(x & y & (1-z)) | x for x in [0, 1], y in [0, 1], z in [0, 1]])
     res = reduceto(SpinGlass{<:SimpleGraph}, CircuitSAT(circuit))
@@ -95,8 +95,8 @@ end
     end
     circuitsat = CircuitSAT(circuit)
     result = reduceto(SpinGlass{<:SimpleGraph}, circuitsat)
-    indexof(x) = findfirst(==(x), circuitsat.symbols[sortperm(result.variables)])
-    gadget = LogicGadget(result.spinglass, indexof.([:x, :y, :z]), [indexof(:d)])
+    indexof2(x) = findfirst(==(x), circuitsat.symbols[sortperm(result.variables)])
+    gadget = LogicGadget(result.spinglass, indexof2.([:x, :y, :z]), [indexof2(:d)])
     tb = truth_table(gadget; variables=circuitsat.symbols[result.variables])
     @test tb.values == vec([((1 - (x & y)) & (1-z)) | x for x in [0, 1], y in [0, 1], z in [0, 1]])
-end
\ No newline at end of file
+end
diff --git a/test/topology.jl b/test/topology.jl
index 87858349..bff02d96 100644
--- a/test/topology.jl
+++ b/test/topology.jl
@@ -40,4 +40,19 @@ end
     @test SimpleEdge(2, 3) in edges(gg)
     @test collect(edges(gg)) == [SimpleEdge(1, 2), SimpleEdge(2, 3)]
     @test vertices(gg) == 1:3
-end
\ No newline at end of file
+
+    grid = GridGraph([(2, 3), (2, 4), (5, 5)], 1.2)
+    g = SimpleGraph(grid)
+    @test ne(g) == 1
+    @test vertices(grid) == vertices(g)
+    @test neighbors(grid, 2) == neighbors(g, 2)
+
+    grid = GridGraph([(2, 3), (2, 4), (5, 5)], 4.0)
+    g = SimpleGraph(grid)
+    @test ne(g) == 3
+    @test vertices(grid) == vertices(g)
+    @test neighbors(grid, 2) == neighbors(g, 2)
+
+    ig, _ = induced_subgraph(grid, [1, 2])
+    @test ne(ig) == 1
+end