Implement isdigraphical and fix isgraphical (#186)

* Make `isgraphical` more robust and add `isdigraphical``

Co-Authored-By: Claudio Moroni <[email protected]>
Co-Authored-By: Pietro Monticone <[email protected]>

Author: 3 people

src/Graphs.jl

@@ -211,6 +211,7 @@ export
     neighborhood,
     neighborhood_dists,
     isgraphical,
+    isdigraphical,
 
     # cycles
     simplecycles_hawick_james,

src/connectivity.jl

@@ -728,17 +728,28 @@ According to Erdös-Gallai theorem, a degree sequence ``\\{d_1, ...,d_n\\}`` (so
 ```math
 \\sum_{i=1}^{r} d_i \\leq r(r-1) + \\sum_{i=r+1}^n min(r,d_i)
 ```
-for each integer r <= n-1
+for each integer r <= n-1. 
+
+See also: [`isdigraphical`](@ref)
 """
-function isgraphical(degs::Vector{<:Integer})
+function isgraphical(degs::AbstractVector{<:Integer})
+    # Check whether the degree sequence is empty
+    !isempty(degs) || return true
+    # Check whether the sum of degrees is even
     iseven(sum(degs)) || return false
+    # Check that all degrees are non negative and less than n-1
+    n = length(degs)
+    all(0 .<= degs .<= n - 1) || return false
+    # Sort the degree sequence in non-increasing order
     sorted_degs = sort(degs; rev=true)
-    n = length(sorted_degs)
+    # Compute the length of the degree sequence
     cur_sum = zero(UInt64)
+    # Compute the minimum of each degree and the corresponding index
     mindeg = Vector{UInt64}(undef, n)
     @inbounds for i in 1:n
         mindeg[i] = min(i, sorted_degs[i])
     end
+    # Check if the degree sequence satisfies the Erdös-Gallai condition
     cum_min = sum(mindeg)
     @inbounds for r in 1:(n - 1)
         cur_sum += sorted_degs[r]
@@ -748,3 +759,67 @@ function isgraphical(degs::Vector{<:Integer})
     end
     return true
 end
+
+"""
+    isdigraphical(indegree_sequence, outdegree_sequence)
+
+Check whether the given indegree sequence and outdegree sequence are digraphical, that is whether they can be the indegree and outdegree sequence of a simple digraph (i.e. a directed graph with no loops). This implies that `indegree_sequence` and `outdegree_sequence` are not independent, as their elements respectively represent the indegrees and outdegrees that the vertices shall have.
+
+### Implementation Notes
+According to Fulkerson-Chen-Anstee theorem, a sequence ``\\{(a_1, b_1), ...,(a_n, b_n)\\}`` (sorted in descending order of a) is graphic iff ``\\sum_{i = 1}^{n} a_i = \\sum_{i = 1}^{n} b_i\\}`` and the sequence obeys the property -
+```math
+\\sum_{i=1}^{r} a_i \\leq \\sum_{i=1}^n min(r-1,b_i) + \\sum_{i=r+1}^n min(r,b_i)
+```
+for each integer 1 <= r <= n-1. 
+
+See also: [`isgraphical`](@ref)
+"""
+function isdigraphical(
+    indegree_sequence::AbstractVector{<:Integer},
+    outdegree_sequence::AbstractVector{<:Integer},
+)
+    # Check whether the degree sequences have the same length 
+    n = length(indegree_sequence)
+    n == length(outdegree_sequence) || throw(
+        ArgumentError("The indegree and outdegree sequences must have the same length.")
+    )
+    # Check whether the degree sequence is empty
+    !(isempty(indegree_sequence) && isempty(outdegree_sequence)) || return true
+    # Check all degrees are non negative and less than n-1
+    all(0 .<= indegree_sequence .<= n - 1) || return false
+    all(0 .<= outdegree_sequence .<= n - 1) || return false
+
+    sum(indegree_sequence) == sum(outdegree_sequence) || return false
+
+    _sortperm = sortperm(indegree_sequence; rev=true)
+
+    sorted_indegree_sequence = indegree_sequence[_sortperm]
+    sorted_outdegree_sequence = outdegree_sequence[_sortperm]
+
+    indegree_sum = zero(Int64)
+    outdegree_min_sum = zero(Int64)
+
+    cum_min = zero(Int64)
+
+    # The following approach, which requires substituting the line
+    # cum_min = sum([min(sorted_outdegree_sequence[i], r) for i in (1+r):n])
+    # with the line
+    # cum_min -= mindeg[r]
+    # inside the for loop below, work as well, but the values of `cum_min` at each iteration differ. To be on the safe side we implemented it as in https://en.wikipedia.org/wiki/Fulkerson%E2%80%93Chen%E2%80%93Anstee_theorem
+    #=     mindeg = Vector{Int64}(undef, n)
+        @inbounds for i = 1:n
+            mindeg[i] = min(i, sorted_outdegree_sequence[i])
+        end
+        cum_min = sum(mindeg) =#
+    # Similarly for `outdegree_min_sum`.
+
+    @inbounds for r in 1:n
+        indegree_sum += sorted_indegree_sequence[r]
+        outdegree_min_sum = sum([min(sorted_outdegree_sequence[i], r - 1) for i in 1:r])
+        cum_min = sum([min(sorted_outdegree_sequence[i], r) for i in (1 + r):n])
+        cond = indegree_sum <= (outdegree_min_sum + cum_min)
+        cond || return false
+    end
+
+    return true
+end

test/connectivity.jl

@@ -297,6 +297,21 @@
     @test @inferred(!isgraphical([1, 1, 1]))
     @test @inferred(isgraphical([2, 2, 2]))
     @test @inferred(isgraphical(fill(3, 10)))
+    @test @inferred(isgraphical(Integer[]))
+    ##@test !@inferred(isgraphical([2]))
+
+    # Test simple digraphicality
+    sdg = SimpleDiGraph(10, 90)
+    @test @inferred(isdigraphical(indegree(sdg), outdegree(sdg)))
+    @test !@inferred(isdigraphical([1, 1, 1], [1, 1, 0]))
+    @test @inferred(isdigraphical(Integer[], Integer[]))
+    #@test !@inferred(isdigraphical([1], [1]))
+    # Self loops should be excluded
+    @test !@inferred(isdigraphical([1], [1]))
+    @test !@inferred(isdigraphical([1, 0], [1, 0]))
+    # Multi edges should be excluded
+    @test !@inferred(isdigraphical([5], [5]))
+
     # 1116
     gc = cycle_graph(4)
     for g in testgraphs(gc)