Stop allocating induced subgraphs

Originally mirroring the networkx implementation, we created a new AbstractGraph object every time we tested subsequent neighbors in the potential PEO with the induced_subgraph function. This commit makes our implementation more performant by simply taking the vertex (sub)set and checking to see if all pairs are adjacent via iteration.

Additionally, we change inconsistent naming in certain parts ('node' vs 'vertex'), changing everything to 'vertex' where relevant.

Author: Luis-Varona

src/chordality.jl

@@ -4,7 +4,7 @@
 Check whether a graph is chordal.
 
 A graph is said to be *chordal* if every cycle of length `≥ 4` has a chord
-(i.e., an edge between two nodes not adjacent in the cycle).
+(i.e., an edge between two vertices not adjacent in the cycle).
 
 ### Performance
 This algorithm is linear in the number of vertices and edges of the graph (i.e.,
@@ -47,32 +47,36 @@ function is_chordal(g::AbstractGraph)
     numbered = Set(start_vertex)
 
     #= Searching by maximum cardinality ensures that in any possible perfect elimination
-    ordering of `g`, `purported_clique_nodes` is precisely the set of neighbors of `v` that
-    come later in the ordering. Hence, if the subgraph induced by `purported_clique_nodes`
+    ordering of `g`, `subsequent_neighbors` is precisely the set of neighbors of `v` that
+    come later in the ordering. Therefore, if the subgraph induced by `subsequent_neighbors`
     in any iteration is not complete, `g` cannot be chordal. =#
     while !isempty(unnumbered)
         # `v` is the vertex in `unnumbered` with the most neighbors in `numbered`
-        v = _max_cardinality_node(g, unnumbered, numbered)
+        v = _max_cardinality_vertex(g, unnumbered, numbered)
         delete!(unnumbered, v)
         push!(numbered, v)
+        subsequent_neighbors = intersect(neighbors(g, v), numbered)
 
-        # A complete subgraph of a larger graph is called a "clique," hence the naming here
-        purported_clique_nodes = intersect(neighbors(g, v), numbered)
-        purported_clique = induced_subgraph(g, purported_clique_nodes)
-
-        _is_complete_graph(purported_clique) || return false
+        # A complete subgraph is also called a "clique," hence the naming here
+        _induces_clique(subsequent_neighbors, g) || return false
     end
 
-    #= That `g` admits a perfect elimination ordering is an "if and only if" condition for
-    chordality, so if every `purported_clique` was indeed complete, `g` must be chordal. =#
+    #= A perfect elimination ordering is an "if and only if" condition for chordality, so if
+    every `subsequent_neighbors` set induced a complete subgraph, `g` must be chordal. =#
     return true
 end
 
-function _max_cardinality_node(
-    g::AbstractGraph, unnumbered::Set{T}, numbered::Set{T}
+function _max_cardinality_vertex(
+    g::AbstractGraph{T}, unnumbered::Set{T}, numbered::Set{T}
 ) where {T}
     cardinality(v::T) = count(in(numbered), neighbors(g, v))
     return argmax(cardinality, unnumbered)
 end
 
-_is_complete_graph(g::AbstractGraph) = density(g) == 1
+function _induces_clique(vertex_subset::Vector{T}, g::AbstractGraph{T}) where {T}
+    for (i, u) in enumerate(vertex_subset), v in Iterators.drop(vertex_subset, i)
+        has_edge(g, u, v) || return false
+    end
+
+    return true
+end