Eulerian cycles/trails for undirected graphs (#232)

* add `eulerian` to compute Eulerian cycles/tours for undirected graphs

* simplify API: don't allow end vertex as input

* actually include new tests in "test/runtests.jl"

* nit

* Doc nit

* fix `@test_throws` use on <v1.8

* fix^N

* code review

docs/src/algorithms/traversals.md

@@ -20,5 +20,6 @@ Pages   = [
     "traversals/greedy_color.jl",
     "traversals/maxadjvisit.jl",
     "traversals/randomwalks.jl",
+    "traversals/eulerian.jl",
 ]
 ```

src/Graphs.jl

@@ -194,6 +194,9 @@ export
     diffusion,
     diffusion_rate,
 
+    # eulerian
+    eulerian,
+
     # coloring
     greedy_color,
 
@@ -488,6 +491,7 @@ include("traversals/dfs.jl")
 include("traversals/maxadjvisit.jl")
 include("traversals/randomwalks.jl")
 include("traversals/diffusion.jl")
+include("traversals/eulerian.jl")
 include("connectivity.jl")
 include("distance.jl")
 include("editdist.jl")

src/Test/Test.jl

@@ -36,6 +36,10 @@ struct GenericGraph{T} <: Graphs.AbstractGraph{T}
     g::SimpleGraph{T}
 end
 
+function GenericGraph(elist::Vector{Graphs.SimpleGraphEdge{T}}) where {T<:Integer}
+    GenericGraph{T}(SimpleGraph(elist))
+end
+
 """
     GenericDiGraph{T} <: Graphs.AbstractGraph{T}
 
@@ -46,6 +50,10 @@ struct GenericDiGraph{T} <: Graphs.AbstractGraph{T}
     g::SimpleDiGraph{T}
 end
 
+function GenericDiGraph(elist::Vector{Graphs.SimpleDiGraphEdge{T}}) where {T<:Integer}
+    GenericDiGraph{T}(SimpleDiGraph(elist))
+end
+
 Graphs.is_directed(::Type{<:GenericGraph}) = false
 Graphs.is_directed(::Type{<:GenericDiGraph}) = true
 

src/traversals/eulerian.jl

@@ -0,0 +1,103 @@
+# Adapted from SimpleGraphs.jl [Copyright (c) 2014, Ed Scheinerman]:
+#    https://github.com/scheinerman/SimpleGraphs.jl/blob/master/src/simple_euler.jl
+# Reproduced under the MIT Expat License.
+
+"""
+    eulerian(g::AbstractSimpleGraph{T}[, u::T]) --> T[]
+
+Returns a [Eulerian trail or cycle](https://en.wikipedia.org/wiki/Eulerian_path) through an
+undirected graph `g`, starting at vertex `u`, returning a vector listing the vertices of `g`
+in the order that they are traversed. If no such trail or cycle exists, throws an error.
+
+A Eulerian trail or cycle is a path that visits every edge of `g` exactly once; for a
+cycle, the path starts _and_ ends at vertex `u`.
+
+## Optional arguments
+- If `u` is omitted, a Eulerian trail or cycle is computed with `u = first(vertices(g))`.
+"""
+function eulerian(g::AbstractGraph{T}, u::T=first(vertices(g))) where {T}
+    is_directed(g) && error("`eulerian` is not yet implemented for directed graphs")
+
+    _check_eulerian_input(g, u) # perform basic sanity checks
+
+    g′ = SimpleGraph{T}(nv(g)) # copy `g` (mutated in `_eulerian!`)
+    for e in edges(g)
+        add_edge!(g′, src(e), dst(e))
+    end
+
+    return _eulerian!(g′, u)
+end
+
+@traitfn function _eulerian!(g::AG::(!IsDirected), u::T) where {T, AG<:AbstractGraph{T}}
+    # TODO: This uses Fleury's algorithm which is O(|E|²) in the number of edges |E|.
+    #       Hierholzer's algorithm [https://en.wikipedia.org/wiki/Eulerian_path#Hierholzer's_algorithm]
+    #       is presumably faster, running in O(|E|) time, but requires needing to keep track
+    #       of visited/nonvisited sites in a doubly-linked list/deque.
+    trail = T[]
+
+    nverts = nv(g)
+    while true
+        # if last vertex
+        if nverts == 1
+            push!(trail, u)
+            return trail
+        end
+
+        Nu = neighbors(g, u)
+        if length(Nu) == 1
+            # if only one neighbor, delete and move on
+            w = first(Nu)
+            rem_edge!(g, u, w)
+            nverts -= 1
+            push!(trail, u)
+            u = w
+        elseif length(Nu) == 0
+            error("graph is not connected: a eulerian cycle/trail does not exist")
+        else
+            # otherwise, pick whichever neighbor is not a bridge/cut-edge
+            bs = bridges(g)
+            for w in Nu
+                if all(e -> _excludes_edge(u, w, e), bs)
+                    # not a bridge/cut-edge; add to trail
+                    rem_edge!(g, u, w)
+                    push!(trail, u)
+                    u = w
+                    break
+                end
+            end
+        end
+    end
+    error("unreachable reached")
+end
+
+@inline function _excludes_edge(u, w, e::AbstractEdge)
+    # `true` if `e` is not `Edge(u,w)` or `Edge(w,u)`, otherwise `false`
+    s, d = src(e), dst(e)
+    return !((u == s && w == d) || (u == d && w == s))
+end
+
+function _check_eulerian_input(g, u)
+    if !has_vertex(g, u)
+        error("starting vertex is not in the graph")
+    end
+
+    # special case: if any vertex has degree zero
+    if any(x->degree(g, x) == 0, vertices(g))
+        error("some vertices have degree zero (are isolated) and cannot be reached")
+    end
+
+    # vertex degree checks
+    du = degree(g, u)
+    if iseven(du)     # cycle: start (u) == stop (v) - all nodes must have even degree
+        if any(x -> isodd(degree(g, x)), vertices(g))
+            error("starting vertex has even degree but there are other vertices with odd degree: a eulerian cycle does not exist")
+        end
+    else # isodd(du)  # trail: start (u) != stop (v) - all nodes, except u and v, must have even degree
+        if count(x -> iseven(degree(g, x)), vertices(g)) != 2
+            error("starting vertex has odd degree but the total number of vertices of odd degree is not equal to 2: a eulerian trail does not exist")
+        end
+    end
+
+    # to reduce cost, the graph connectivity check is performed in `_eulerian!` rather
+    # than through `is_connected(g)`
+end

test/runtests.jl

@@ -116,6 +116,7 @@ tests = [
     "traversals/maxadjvisit",
     "traversals/randomwalks",
     "traversals/diffusion",
+    "traversals/eulerian",
     "community/cliques",
     "community/core-periphery",
     "community/label_propagation",

test/traversals/eulerian.jl

@@ -0,0 +1,37 @@
+@testset "Eulerian tours/cycles" begin
+    # a cycle (identical start/end)
+    g0 = GenericGraph([Edge(1,2), Edge(2,3), Edge(3,1)])
+    @test eulerian(g0, 1) == eulerian(g0)
+    @test last(eulerian(g0, 1)) == 1 # a cycle
+
+    # a tour (different start/end)
+    g1 = GenericGraph([Edge(1,2), Edge(2,3), Edge(3,4)])
+    @test eulerian(g1, 1) == [1,2,3,4]
+    @test_throws ErrorException("starting vertex has even degree but there are other vertices with odd degree: a eulerian cycle does not exist") eulerian(g1, 2)
+
+    # a cycle with a node (vertex 2) with multiple neighbors
+    g2 = GenericGraph([Edge(1,2), Edge(2,3), Edge(3,4), Edge(4,1), Edge(2,5), Edge(5,6), 
+                      Edge(6,2)])
+    @test eulerian(g2) == eulerian(g2, 1) == [1, 2, 5, 6, 2, 3, 4, 1]
+
+    # graph with odd-degree vertices
+    g3 = GenericGraph([Edge(1,2), Edge(2,3), Edge(3,4), Edge(2,4), Edge(4,1), Edge(4,2)])
+    @test_throws ErrorException("starting vertex has even degree but there are other vertices with odd degree: a eulerian cycle does not exist") eulerian(g3, 1)
+
+    # start/end point not in graph
+    @test_throws ErrorException("starting vertex is not in the graph") eulerian(g3, 5)
+
+    # disconnected components
+    g4 = GenericGraph([Edge(1,2), Edge(2,3), Edge(3,1),  # component 1
+                      Edge(4,5), Edge(5,6), Edge(6,4)]) # component 2
+    @test_throws ErrorException("graph is not connected: a eulerian cycle/trail does not exist") eulerian(g4)
+
+    # zero-degree nodes
+    g5′ = SimpleGraph(4)
+    add_edge!(g5′, Edge(1,2)); add_edge!(g5′, Edge(2,3)); add_edge!(g5′, Edge(3,1))
+    g5 = GenericGraph(g5′)
+    @test_throws ErrorException("some vertices have degree zero (are isolated) and cannot be reached") eulerian(g5)
+
+    # not yet implemented for directed graphs
+    @test_broken eulerian(GenericDiGraph([Edge(1,2), Edge(2,3), Edge(3,1)]))
+end