def reduction2(g: MultiGraph, w: set, h: MultiGraph, k: int) -> (int, int, bool):
for v in h.nodes():
# Check if G[W ∪ {v}] contains a cycle.
if not is_forest(g.subgraph(w.union({v}))):
g.remove_node(v)
h.remove_nodes_from([v])
return (k - 1, v, True)
return (k, None, False)
def fvs_disjoint(g: MultiGraph, w: set, k: int) -> set:
# Check that G[W] is a forest.
# If it isn't, then a solution X not using W can't remove W's cycles.
if not is_forest(g.subgraph(w)):
return None
# Apply reductions exhaustively.
k, soln_redux = apply_reductions(g, w, k)
# If k becomes negative, it indicates that the reductions included
# more than k vertices, hence no solution of size <= k exists.
if k < 0:
return None
# If G has been reduced to nothing and k is >= 0 then the solution generated by the reductions
# is already optimal.
if len(g) == 0:
return soln_redux
# Find an x in H of degree at most 1.
h = graph_minus(g, w)
x = None
for v in h.nodes():
if h.degree(v) <= 1:
x = v
break
assert x is not None
# Branch.
# G is copied in the left branch (as it is modified), but passed directly in the right.
soln_left = fvs_disjoint(graph_minus(g, {x}), w, k - 1)
if soln_left is not None:
return soln_redux.union(soln_left).union({x})
soln_right = fvs_disjoint(g, w.union({x}), k)
if soln_right is not None:
return soln_redux.union(soln_right)
return None
def is_fvs(g: MultiGraph, w) -> bool: h = graph_minus(g, w) return ((len(h) == 0) or is_forest(h))