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Resolve nice tree decomp bug in #36843, and allow label_nice_tree_decomposition to return a digraph #36846

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merged 5 commits into from
Dec 19, 2023
Merged

Resolve nice tree decomp bug in #36843, and allow label_nice_tree_decomposition to return a digraph #36846

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8b2c666
tree_decomposition.pyx: Allow returning directed nice tree decomp; fi…
guojing0 Dec 9, 2023
270bd48
tree_decomposition.pyx: Add tests for small simple graphs; change doc…
guojing0 Dec 10, 2023
fadce49
Merge branch 'develop' into fix-nice-tree-decomp
guojing0 Dec 10, 2023
157c801
tree_decomposition.pyx: Change K_{1, 4} to claw in label_nice_tree_de…
guojing0 Dec 13, 2023
2b6f2a9
tree_decomposition.pyx: Make test random
guojing0 Dec 13, 2023
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106 changes: 78 additions & 28 deletions src/sage/graphs/graph_decompositions/tree_decomposition.pyx
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Original file line number Diff line number Diff line change
Expand Up @@ -818,6 +818,7 @@ def make_nice_tree_decomposition(graph, tree_decomp):
INPUT:

- ``graph`` -- a Sage graph

- ``tree_decomp`` -- a tree decomposition

OUTPUT:
Expand Down Expand Up @@ -853,6 +854,46 @@ def make_nice_tree_decomposition(graph, tree_decomp):
sage: bip_one_four_TD = bip_one_four.treewidth(certificate=True)
sage: make_nice_tree_decomposition(bip_one_four, bip_one_four_TD)
Nice tree decomposition of Tree decomposition: Graph on 15 vertices

Check that :issue:`36843` is fixed::

sage: from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition
sage: triangle = graphs.CompleteGraph(3)
sage: triangle_TD = triangle.treewidth(certificate=True)
sage: make_nice_tree_decomposition(triangle, triangle_TD)
Nice tree decomposition of Tree decomposition: Graph on 7 vertices

::

sage: from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition
sage: graph = graphs.CompleteBipartiteGraph(2, 5)
sage: graph_TD = graph.treewidth(certificate=True)
sage: make_nice_tree_decomposition(graph, graph_TD)
Nice tree decomposition of Tree decomposition: Graph on 25 vertices
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::

sage: from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition
sage: empty_graph = graphs.EmptyGraph()
sage: tree_decomp = empty_graph.treewidth(certificate=True)
sage: len(make_nice_tree_decomposition(empty_graph, tree_decomp))
0

::

sage: from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition
sage: singleton = graphs.CompleteGraph(1)
sage: tree_decomp = singleton.treewidth(certificate=True)
sage: make_nice_tree_decomposition(singleton, tree_decomp)
Nice tree decomposition of Tree decomposition: Graph on 3 vertices

::

sage: from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition
sage: an_edge = graphs.CompleteGraph(2)
sage: tree_decomp = an_edge.treewidth(certificate=True)
sage: make_nice_tree_decomposition(an_edge, tree_decomp)
Nice tree decomposition of Tree decomposition: Graph on 5 vertices
"""
if not is_valid_tree_decomposition(graph, tree_decomp):
raise ValueError("input must be a valid tree decomposition for this graph")
Expand All @@ -864,11 +905,19 @@ def make_nice_tree_decomposition(graph, tree_decomp):

# Step 1: Ensure the tree is directed and has a root
# Choose a root and orient the edges from root-to-leaves direction
leaves = [u for u in tree_decomp if tree_decomp.degree(u) == 1]
root = leaves.pop()
#
# Testing <= 1 for the special case when one bag containing all vertices
leaves = [u for u in tree_decomp if tree_decomp.degree(u) <= 1]

from sage.graphs.digraph import DiGraph
directed_tree = DiGraph(tree_decomp.breadth_first_search(start=root, edges=True),
format='list_of_edges')
if len(leaves) == 1:
root = leaves[0]
directed_tree = DiGraph(tree_decomp)
else:
root = leaves.pop()

directed_tree = DiGraph(tree_decomp.breadth_first_search(start=root, edges=True),
format='list_of_edges')

# Relabel the graph in range (0, |tree_decomp| - 1)
bags_to_int = directed_tree.relabel(inplace=True, return_map=True)
Expand Down Expand Up @@ -905,7 +954,7 @@ def make_nice_tree_decomposition(graph, tree_decomp):
children = directed_tree.neighbors_out(ui)
children.pop() # one vertex remains a child of ui

directed_tree.delete_edges((ui, vi) for v in children)
directed_tree.delete_edges((ui, vi) for vi in children)

new_nodes = [directed_tree.add_vertex() for _ in range(len(children) - 1)]

Expand Down Expand Up @@ -1012,7 +1061,8 @@ def make_nice_tree_decomposition(graph, tree_decomp):

return nice_tree_decomp

def label_nice_tree_decomposition(nice_TD, root):

def label_nice_tree_decomposition(nice_TD, root, directed=False):
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r"""
Return a nice tree decomposition with nodes labelled accordingly.

Expand All @@ -1022,52 +1072,50 @@ def label_nice_tree_decomposition(nice_TD, root):

- ``root`` -- the root of the nice tree decomposition

- ``directed`` -- boolean (default: ``False``); whether to return the nice
tree decomposition as a directed graph rooted at vertex ``root`` or as an
undirected graph

OUTPUT:

A nice tree decomposition with nodes labelled.

EXAMPLES::

sage: from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition, label_nice_tree_decomposition
sage: bip_one_four = graphs.CompleteBipartiteGraph(1, 4)
sage: bip_one_four_TD = bip_one_four.treewidth(certificate=True)
sage: nice_TD = make_nice_tree_decomposition(bip_one_four, bip_one_four_TD)
sage: claw = graphs.CompleteBipartiteGraph(1, 3)
sage: claw_TD = claw.treewidth(certificate=True)
sage: nice_TD = make_nice_tree_decomposition(claw, claw_TD)
sage: root = sorted(nice_TD.vertices())[0]
sage: label_TD = label_nice_tree_decomposition(nice_TD, root)
sage: for node in sorted(label_TD):
sage: label_TD = label_nice_tree_decomposition(nice_TD, root, directed=True)
sage: label_TD.name()
'Labelled Nice tree decomposition of Tree decomposition'
sage: for node in sorted(label_TD): # random
....: print(node, label_TD.get_vertex(node))
(0, {}) root
(0, {}) forget
(1, {0}) forget
(2, {0, 1}) intro
(3, {0}) forget
(4, {0, 4}) join
(5, {0, 4}) intro
(6, {0, 4}) intro
(7, {0}) forget
(8, {0}) forget
(9, {0, 3}) intro
(10, {0, 2}) intro
(11, {3}) intro
(12, {2}) intro
(13, {}) leaf
(14, {}) leaf
(4, {0, 3}) intro
(5, {0}) forget
(6, {0, 2}) intro
(7, {2}) intro
(8, {}) leaf
"""
from sage.graphs.digraph import DiGraph
from sage.graphs.graph import Graph

directed_TD = DiGraph(nice_TD.breadth_first_search(start=root, edges=True),
format='list_of_edges')
format='list_of_edges',
name='Labelled {}'.format(nice_TD.name()))

# The loop starts from the root node
# We assume the tree decomposition is valid and nice,
# hence saving time on checking.
for node in directed_TD:
in_deg = directed_TD.in_degree(node)
out_deg = directed_TD.out_degree(node)

if in_deg == 0:
directed_TD.set_vertex(node, 'root')
elif out_deg == 2:
if out_deg == 2:
directed_TD.set_vertex(node, 'join')
elif out_deg == 1:
current_bag = node[1]
Expand All @@ -1080,6 +1128,8 @@ def label_nice_tree_decomposition(nice_TD, root):
else:
directed_TD.set_vertex(node, 'leaf')

if directed:
return directed_TD
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return Graph(directed_TD, name=nice_TD.name())


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