Tree decompositions¶
This module implements tree-decomposition methods.
A tree-decomposition of a graph \(G = (V, E)\) is a pair \((X, T)\), where \(X=\{X_1, X_2, \ldots, X_t\}\) is a family of subsets of \(V\), usually called bags, and \(T\) is a tree of order \(t\) whose nodes are the subsets \(X_i\) satisfying the following properties:
The union of all sets \(X_i\) equals \(V\). That is, each vertex of the graph \(G\) is associated with at least one tree node.
For every edge \((v, w)\) in the graph, there is a subset \(X_i\) that contains both \(v\) and \(w\). That is, each edge of the graph \(G\) appears in a tree node.
The nodes associated with vertex \(v \in V\) form a connected subtree of \(T\). That is, if \(X_i\) and \(X_j\) both contain a vertex \(v \in V\), then all nodes \(X_k\) of the tree in the (unique) path between \(X_i\) and \(X_j\) contain \(v\) as well, and we have \(X_i \cap X_j \subseteq X_k\).
The width of a tree decomposition is the size of the largest set \(X_i\) minus one, i.e., \(\max_{X_i \in X} |X_i| - 1\), and the treewidth \(tw(G)\) of a graph \(G\) is the minimum width among all possible tree decompositions of \(G\). Observe that, the size of the largest set is diminished by one in order to make the treewidth of a tree equal to one.
The length of a tree decomposition, as proposed in [DG2006], is the maximum diameter in \(G\) of its bags, where the diameter of a bag \(X_i\) is the largest distance in \(G\) between the vertices in \(X_i\) (i.e., \(\max_{u, v \in X_i} \dist_G(u, v)\)). The treelength \(tl(G)\) of a graph \(G\) is the minimum length among all possible tree decompositions of \(G\).
While deciding whether a graph has treelength 1 can be done in linear time (equivalent to deciding if the graph is chordal), deciding if it has treelength at most \(k\) for any fixed constant \(k \leq 2\) is NP-complete [Lokshtanov2009].
Treewidth and treelength are different measures of tree-likeness. In particular, trees have treewidth and treelength 1:
sage: T = graphs.RandomTree(20)
sage: T.treewidth()
1
sage: T.treelength()
1
>>> from sage.all import *
>>> T = graphs.RandomTree(Integer(20))
>>> T.treewidth()
1
>>> T.treelength()
1
The treewidth of a cycle is 2 and its treelength is \(\lceil n/3 \rceil\):
sage: [graphs.CycleGraph(n).treewidth() for n in range(3, 11)]
[2, 2, 2, 2, 2, 2, 2, 2]
sage: [graphs.CycleGraph(n).treelength() for n in range(3, 11)]
[1, 2, 2, 2, 3, 3, 3, 4]
>>> from sage.all import *
>>> [graphs.CycleGraph(n).treewidth() for n in range(Integer(3), Integer(11))]
[2, 2, 2, 2, 2, 2, 2, 2]
>>> [graphs.CycleGraph(n).treelength() for n in range(Integer(3), Integer(11))]
[1, 2, 2, 2, 3, 3, 3, 4]
The treewidth of a clique is \(n-1\) and its treelength is 1:
sage: [graphs.CompleteGraph(n).treewidth() for n in range(3, 11)]
[2, 3, 4, 5, 6, 7, 8, 9]
sage: [graphs.CompleteGraph(n).treelength() for n in range(3, 11)]
[1, 1, 1, 1, 1, 1, 1, 1]
>>> from sage.all import *
>>> [graphs.CompleteGraph(n).treewidth() for n in range(Integer(3), Integer(11))]
[2, 3, 4, 5, 6, 7, 8, 9]
>>> [graphs.CompleteGraph(n).treelength() for n in range(Integer(3), Integer(11))]
[1, 1, 1, 1, 1, 1, 1, 1]
This module contains the following methods
Compute the treewidth of \(G\) (and provide a decomposition). |
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Compute the treelength of \(G\) (and provide a decomposition). |
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Return a nice tree decomposition (TD) of the TD |
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Return a nice tree decomposition with nodes labelled accordingly. |
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Check whether \(T\) is a valid tree-decomposition for \(G\). |
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Return a reduced tree-decomposition of \(T\). |
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Return the width of the tree decomposition \(T\) of \(G\). |
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Return the length of the tree decomposition \(T\) of \(G\). |
Methods¶
- class sage.graphs.graph_decompositions.tree_decomposition.TreelengthConnected[source]¶
Bases:
objectCompute the treelength of a connected graph (and provide a decomposition).
This class implements an algorithm for computing the treelength of a connected graph that virtually explores the graph of all pairs
(vertex_cut, connected_component), wherevertex_cutis a vertex cut of the graph of length \(\leq k\), andconnected_componentis a connected component of the graph induced byG - vertex_cut.We deduce that the pair
(vertex_cut, connected_component)is feasible with treelength \(k\) ifconnected_componentis empty, or if a vertexvfromvertex_cutcan be replaced with a vertex fromconnected_component, such that the pair(vertex_cut + v, connected_component - v)is feasible.INPUT:
G– a sage Graphk– integer (default:None); indicates the length to be considered. When \(k\) is an integer, the method checks that the graph has treelength \(\leq k\). If \(k\) isNone(default), the method computes the optimal treelength.certificate– boolean (default:False); whether to also compute the tree-decomposition itself
OUTPUT:
TreelengthConnected(G)returns the treelength of \(G\). When \(k\) is specified, it returnsFalsewhen no tree-decomposition of length \(\leq k\) exists orTrueotherwise. Whencertificate=True, the tree-decomposition is also returned.EXAMPLES:
A clique has treelength 1:
sage: from sage.graphs.graph_decompositions.tree_decomposition import TreelengthConnected sage: TreelengthConnected(graphs.CompleteGraph(3)).get_length() 1 sage: TC = TreelengthConnected(graphs.CompleteGraph(4), certificate=True) sage: TC.get_length() 1 sage: TC.get_tree_decomposition() Tree decomposition of Complete graph: Graph on 1 vertex
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import TreelengthConnected >>> TreelengthConnected(graphs.CompleteGraph(Integer(3))).get_length() 1 >>> TC = TreelengthConnected(graphs.CompleteGraph(Integer(4)), certificate=True) >>> TC.get_length() 1 >>> TC.get_tree_decomposition() Tree decomposition of Complete graph: Graph on 1 vertex
A cycle has treelength \(\lceil n/3 \rceil\):
sage: TreelengthConnected(graphs.CycleGraph(6)).get_length() 2 sage: TreelengthConnected(graphs.CycleGraph(7)).get_length() 3 sage: TreelengthConnected(graphs.CycleGraph(7), k=3).is_less_than_k() True sage: TreelengthConnected(graphs.CycleGraph(7), k=2).is_less_than_k() False
>>> from sage.all import * >>> TreelengthConnected(graphs.CycleGraph(Integer(6))).get_length() 2 >>> TreelengthConnected(graphs.CycleGraph(Integer(7))).get_length() 3 >>> TreelengthConnected(graphs.CycleGraph(Integer(7)), k=Integer(3)).is_less_than_k() True >>> TreelengthConnected(graphs.CycleGraph(Integer(7)), k=Integer(2)).is_less_than_k() False
- get_length()[source]¶
Return the length of the tree decomposition.
EXAMPLES:
sage: from sage.graphs.graph_decompositions.tree_decomposition import TreelengthConnected sage: G = graphs.CycleGraph(4) sage: TreelengthConnected(G).get_length() 2 sage: TreelengthConnected(G, k=2).get_length() 2 sage: TreelengthConnected(G, k=1).get_length() Traceback (most recent call last): ... ValueError: no tree decomposition with length <= 1 was found
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import TreelengthConnected >>> G = graphs.CycleGraph(Integer(4)) >>> TreelengthConnected(G).get_length() 2 >>> TreelengthConnected(G, k=Integer(2)).get_length() 2 >>> TreelengthConnected(G, k=Integer(1)).get_length() Traceback (most recent call last): ... ValueError: no tree decomposition with length <= 1 was found
- get_tree_decomposition()[source]¶
Return the tree-decomposition.
EXAMPLES:
sage: from sage.graphs.graph_decompositions.tree_decomposition import TreelengthConnected sage: G = graphs.CycleGraph(4) sage: TreelengthConnected(G, certificate=True).get_tree_decomposition() Tree decomposition of Cycle graph: Graph on 2 vertices sage: G.diameter() 2 sage: TreelengthConnected(G, k=2, certificate=True).get_tree_decomposition() Tree decomposition of Cycle graph: Graph on 1 vertex sage: TreelengthConnected(G, k=1, certificate=True).get_tree_decomposition() Traceback (most recent call last): ... ValueError: no tree decomposition with length <= 1 was found
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import TreelengthConnected >>> G = graphs.CycleGraph(Integer(4)) >>> TreelengthConnected(G, certificate=True).get_tree_decomposition() Tree decomposition of Cycle graph: Graph on 2 vertices >>> G.diameter() 2 >>> TreelengthConnected(G, k=Integer(2), certificate=True).get_tree_decomposition() Tree decomposition of Cycle graph: Graph on 1 vertex >>> TreelengthConnected(G, k=Integer(1), certificate=True).get_tree_decomposition() Traceback (most recent call last): ... ValueError: no tree decomposition with length <= 1 was found
- is_less_than_k()[source]¶
Return whether a tree decomposition with length at most \(k\) was found.
EXAMPLES:
sage: from sage.graphs.graph_decompositions.tree_decomposition import TreelengthConnected sage: G = graphs.CycleGraph(4) sage: TreelengthConnected(G, k=1).is_less_than_k() False sage: TreelengthConnected(G, k=2).is_less_than_k() True sage: TreelengthConnected(G).is_less_than_k() Traceback (most recent call last): ... ValueError: parameter 'k' has not been specified
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import TreelengthConnected >>> G = graphs.CycleGraph(Integer(4)) >>> TreelengthConnected(G, k=Integer(1)).is_less_than_k() False >>> TreelengthConnected(G, k=Integer(2)).is_less_than_k() True >>> TreelengthConnected(G).is_less_than_k() Traceback (most recent call last): ... ValueError: parameter 'k' has not been specified
- sage.graphs.graph_decompositions.tree_decomposition.is_valid_tree_decomposition(G, T)[source]¶
Check whether \(T\) is a valid tree-decomposition for \(G\).
INPUT:
G– a sage GraphT– a tree decomposition, i.e., a tree whose vertices are the bags (subsets of vertices) of the decomposition
EXAMPLES:
sage: from sage.graphs.graph_decompositions.tree_decomposition import is_valid_tree_decomposition sage: K = graphs.CompleteGraph(4) sage: T = Graph() sage: T.add_vertex(Set(K)) sage: is_valid_tree_decomposition(K, T) True sage: G = graphs.RandomGNP(10, .2) sage: T = G.treewidth(certificate=True) sage: is_valid_tree_decomposition(G, T) True
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import is_valid_tree_decomposition >>> K = graphs.CompleteGraph(Integer(4)) >>> T = Graph() >>> T.add_vertex(Set(K)) >>> is_valid_tree_decomposition(K, T) True >>> G = graphs.RandomGNP(Integer(10), RealNumber('.2')) >>> T = G.treewidth(certificate=True) >>> is_valid_tree_decomposition(G, T) True
The union of the bags is the set of vertices of \(G\):
sage: G = graphs.PathGraph(4) sage: T = G.treewidth(certificate=True) sage: _ = G.add_vertex() sage: is_valid_tree_decomposition(G, T) False
>>> from sage.all import * >>> G = graphs.PathGraph(Integer(4)) >>> T = G.treewidth(certificate=True) >>> _ = G.add_vertex() >>> is_valid_tree_decomposition(G, T) False
Each edge of \(G\) is contained in a bag:
sage: G = graphs.PathGraph(4) sage: T = G.treewidth(certificate=True) sage: G.add_edge(0, 3) sage: is_valid_tree_decomposition(G, T) False
>>> from sage.all import * >>> G = graphs.PathGraph(Integer(4)) >>> T = G.treewidth(certificate=True) >>> G.add_edge(Integer(0), Integer(3)) >>> is_valid_tree_decomposition(G, T) False
The bags containing a vertex \(v\) form a subtree of \(T\):
sage: G = graphs.PathGraph(4) sage: X1, X2, X3 = Set([0, 1]), Set([1, 2]), Set([2, 3]) sage: T = Graph([(X1, X3), (X3, X2)]) sage: is_valid_tree_decomposition(G, T) False
>>> from sage.all import * >>> G = graphs.PathGraph(Integer(4)) >>> X1, X2, X3 = Set([Integer(0), Integer(1)]), Set([Integer(1), Integer(2)]), Set([Integer(2), Integer(3)]) >>> T = Graph([(X1, X3), (X3, X2)]) >>> is_valid_tree_decomposition(G, T) False
- sage.graphs.graph_decompositions.tree_decomposition.label_nice_tree_decomposition(nice_TD, root, directed=False)[source]¶
Return a nice tree decomposition with nodes labelled accordingly.
INPUT:
nice_TD– a nice tree decompositionroot– the root of the nice tree decompositiondirected– boolean (default:False); whether to return the nice tree decomposition as a directed graph rooted at vertexrootor 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: 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, 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, {}) forget (1, {0}) forget (2, {0, 1}) intro (3, {0}) forget (4, {0, 3}) intro (5, {0}) forget (6, {0, 2}) intro (7, {2}) intro (8, {}) leaf
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition, label_nice_tree_decomposition >>> claw = graphs.CompleteBipartiteGraph(Integer(1), Integer(3)) >>> claw_TD = claw.treewidth(certificate=True) >>> nice_TD = make_nice_tree_decomposition(claw, claw_TD) >>> root = sorted(nice_TD.vertices())[Integer(0)] >>> label_TD = label_nice_tree_decomposition(nice_TD, root, directed=True) >>> label_TD.name() 'Labelled Nice tree decomposition of Tree decomposition' >>> for node in sorted(label_TD): # random ... print(node, label_TD.get_vertex(node)) (0, {}) forget (1, {0}) forget (2, {0, 1}) intro (3, {0}) forget (4, {0, 3}) intro (5, {0}) forget (6, {0, 2}) intro (7, {2}) intro (8, {}) leaf
- sage.graphs.graph_decompositions.tree_decomposition.length_of_tree_decomposition(G, T, check=True)[source]¶
Return the length of the tree decomposition \(T\) of \(G\).
The length of a tree decomposition, as proposed in [DG2006], is the maximum diameter in \(G\) of its bags, where the diameter of a bag \(X_i\) is the largest distance in \(G\) between the vertices in \(X_i\) (i.e., \(\max_{u, v \in X_i} \dist_G(u, v)\)). See the documentation of the
tree_decompositionmodule for more details.INPUT:
G– a graphT– a tree-decomposition for \(G\)check– boolean (default:True); whether to check that the tree-decomposition \(T\) is valid for \(G\)
EXAMPLES:
Trees and cliques have treelength 1:
sage: from sage.graphs.graph_decompositions.tree_decomposition import length_of_tree_decomposition sage: G = graphs.CompleteGraph(5) sage: tl, T = G.treelength(certificate=True) sage: tl 1 sage: length_of_tree_decomposition(G, T, check=True) 1 sage: G = graphs.RandomTree(20) sage: tl, T = G.treelength(certificate=True) sage: tl 1 sage: length_of_tree_decomposition(G, T, check=True) 1
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import length_of_tree_decomposition >>> G = graphs.CompleteGraph(Integer(5)) >>> tl, T = G.treelength(certificate=True) >>> tl 1 >>> length_of_tree_decomposition(G, T, check=True) 1 >>> G = graphs.RandomTree(Integer(20)) >>> tl, T = G.treelength(certificate=True) >>> tl 1 >>> length_of_tree_decomposition(G, T, check=True) 1
The Petersen graph has treelength 2:
sage: G = graphs.PetersenGraph() sage: tl, T = G.treelength(certificate=True) sage: tl 2 sage: length_of_tree_decomposition(G, T) 2
>>> from sage.all import * >>> G = graphs.PetersenGraph() >>> tl, T = G.treelength(certificate=True) >>> tl 2 >>> length_of_tree_decomposition(G, T) 2
When a tree-decomposition has a single bag containing all vertices of a graph, the length of this tree-decomposition is the diameter of the graph:
sage: G = graphs.Grid2dGraph(2, 5) sage: G.treelength() 2 sage: G.diameter() 5 sage: T = Graph({Set(G): []}) sage: length_of_tree_decomposition(G, T) 5
>>> from sage.all import * >>> G = graphs.Grid2dGraph(Integer(2), Integer(5)) >>> G.treelength() 2 >>> G.diameter() 5 >>> T = Graph({Set(G): []}) >>> length_of_tree_decomposition(G, T) 5
- sage.graphs.graph_decompositions.tree_decomposition.make_nice_tree_decomposition(graph, tree_decomp)[source]¶
Return a nice tree decomposition (TD) of the TD
tree_decomp.See page 161 of [CFKLMPPS15] for a description of the nice tree decomposition.
A nice TD \(NT\) is a rooted tree with four types of nodes:
Leaf nodes have no children and bag size 1;
Introduce nodes have one child: If \(v \in NT\) is an introduce node and \(w \in NT\) its child, then \(Bag(v) = Bag(w) \cup \{ x \}\), where \(x\) is the introduced node;
Forget nodes have one child: If \(v \in NT\) is a forget node and \(w \in NT\) its child, then \(Bag(v) = Bag(w) \setminus \{ x \}\), where \(x\) is the forgotten node;
Join nodes have two children, both identical to the parent.
INPUT:
graph– a Sage graphtree_decomp– a tree decomposition
OUTPUT: a nice tree decomposition
Warning
This method assumes that the vertices of the input tree
tree_decompare hashable and have attributeissuperset, e.g.,frozensetorSet_object_enumerated_with_category.EXAMPLES:
sage: from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition sage: petersen = graphs.PetersenGraph() sage: petersen_TD = petersen.treewidth(certificate=True) sage: make_nice_tree_decomposition(petersen, petersen_TD) Nice tree decomposition of Tree decomposition: Graph on 28 vertices
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition >>> petersen = graphs.PetersenGraph() >>> petersen_TD = petersen.treewidth(certificate=True) >>> make_nice_tree_decomposition(petersen, petersen_TD) Nice tree decomposition of Tree decomposition: Graph on 28 vertices
sage: from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition sage: cherry = graphs.CompleteBipartiteGraph(1, 2) sage: cherry_TD = cherry.treewidth(certificate=True) sage: make_nice_tree_decomposition(cherry, cherry_TD) Nice tree decomposition of Tree decomposition: Graph on 7 vertices
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition >>> cherry = graphs.CompleteBipartiteGraph(Integer(1), Integer(2)) >>> cherry_TD = cherry.treewidth(certificate=True) >>> make_nice_tree_decomposition(cherry, cherry_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: bip_one_four = graphs.CompleteBipartiteGraph(1, 4) 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
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition >>> bip_one_four = graphs.CompleteBipartiteGraph(Integer(1), Integer(4)) >>> bip_one_four_TD = bip_one_four.treewidth(certificate=True) >>> 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
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition >>> triangle = graphs.CompleteGraph(Integer(3)) >>> triangle_TD = triangle.treewidth(certificate=True) >>> 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
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition >>> graph = graphs.CompleteBipartiteGraph(Integer(2), Integer(5)) >>> graph_TD = graph.treewidth(certificate=True) >>> make_nice_tree_decomposition(graph, graph_TD) Nice tree decomposition of Tree decomposition: Graph on 25 vertices
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
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition >>> empty_graph = graphs.EmptyGraph() >>> tree_decomp = empty_graph.treewidth(certificate=True) >>> 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
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition >>> singleton = graphs.CompleteGraph(Integer(1)) >>> tree_decomp = singleton.treewidth(certificate=True) >>> 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
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import make_nice_tree_decomposition >>> an_edge = graphs.CompleteGraph(Integer(2)) >>> tree_decomp = an_edge.treewidth(certificate=True) >>> make_nice_tree_decomposition(an_edge, tree_decomp) Nice tree decomposition of Tree decomposition: Graph on 5 vertices
- sage.graphs.graph_decompositions.tree_decomposition.reduced_tree_decomposition(T)[source]¶
Return a reduced tree-decomposition of \(T\).
We merge all edges between two sets \(S\) and \(S'\) where \(S\) is a subset of \(S'\). To do so, we use a simple union-find data structure to record merge operations and the good sets.
Warning
This method assumes that the vertices of the input tree \(T\) are hashable and have attribute
issuperset, e.g.,frozensetorSet_object_enumerated.INPUT:
T– a tree-decomposition
EXAMPLES:
sage: from sage.graphs.graph_decompositions.tree_decomposition import reduced_tree_decomposition sage: from sage.graphs.graph_decompositions.tree_decomposition import is_valid_tree_decomposition sage: G = graphs.PathGraph(3) sage: T = Graph() sage: T.add_path([Set([0]), Set([0, 1]), Set([1]), Set([1, 2]), Set([2])]) sage: T.order() 5 sage: is_valid_tree_decomposition(G, T) True sage: T2 = reduced_tree_decomposition(T) sage: is_valid_tree_decomposition(G, T2) True sage: T2.order() 2
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import reduced_tree_decomposition >>> from sage.graphs.graph_decompositions.tree_decomposition import is_valid_tree_decomposition >>> G = graphs.PathGraph(Integer(3)) >>> T = Graph() >>> T.add_path([Set([Integer(0)]), Set([Integer(0), Integer(1)]), Set([Integer(1)]), Set([Integer(1), Integer(2)]), Set([Integer(2)])]) >>> T.order() 5 >>> is_valid_tree_decomposition(G, T) True >>> T2 = reduced_tree_decomposition(T) >>> is_valid_tree_decomposition(G, T2) True >>> T2.order() 2
- sage.graphs.graph_decompositions.tree_decomposition.treelength(G, k=None, certificate=False)[source]¶
Compute the treelength of \(G\) (and provide a decomposition).
The length of a tree decomposition, as proposed in [DG2006], is the maximum diameter in \(G\) of its bags, where the diameter of a bag \(X_i\) is the largest distance in \(G\) between the vertices in \(X_i\) (i.e., \(\max_{u, v \in X_i} \dist_G(u, v)\)). The treelength \(tl(G)\) of a graph \(G\) is the minimum length among all possible tree decompositions of \(G\). See the documentation of the
tree_decompositionmodule for more details.INPUT:
G– a sage Graphk– integer (default:None); indicates the length to be considered. When \(k\) is an integer, the method checks that the graph has treelength \(\leq k\). If \(k\) isNone(default), the method computes the optimal treelength.certificate– boolean (default:False); whether to also return the tree-decomposition itself
OUTPUT:
G.treelength()returns the treelength of \(G\). When \(k\) is specified, it returnsFalsewhen no tree-decomposition of length \(\leq k\) exists orTrueotherwise. Whencertificate=True, the tree-decomposition is also returned.ALGORITHM:
This method virtually explores the graph of all pairs
(vertex_cut, connected_component), wherevertex_cutis a vertex cut of the graph of length \(\leq k\), andconnected_componentis a connected component of the graph induced byG - vertex_cut.We deduce that the pair
(vertex_cut, connected_component)is feasible with treelength \(k\) ifconnected_componentis empty, or if a vertexvfromvertex_cutcan be replaced with a vertex fromconnected_component, such that the pair(vertex_cut + v, connected_component - v)is feasible.In practice, this method decomposes the graph by its clique minimal separators into atoms, computes the treelength of each of atom and returns the maximum value over all the atoms. Indeed, we have that \(tl(G) = \max_{X \in A} tl(G[X])\) where \(A\) is the set of atoms of the decomposition by clique separators of \(G\). When
certificate == True, the tree-decompositions of the atoms are connected to each others by adding edges with respect to the clique separators.See also
treewidth()computes the treewidth of a graph.path_decomposition()computes the pathwidth of a graph.module
vertex_separation.
EXAMPLES:
The PetersenGraph has treelength 2:
sage: G = graphs.PetersenGraph() sage: G.treelength() 2
>>> from sage.all import * >>> G = graphs.PetersenGraph() >>> G.treelength() 2
Disconnected graphs have infinite treelength:
sage: G = Graph(2) sage: G.treelength() +Infinity sage: G.treelength(k=+Infinity) True sage: G.treelength(k=2) False sage: G.treelength(certificate=True) Traceback (most recent call last): ... ValueError: the tree decomposition of a disconnected graph is not defined
>>> from sage.all import * >>> G = Graph(Integer(2)) >>> G.treelength() +Infinity >>> G.treelength(k=+Infinity) True >>> G.treelength(k=Integer(2)) False >>> G.treelength(certificate=True) Traceback (most recent call last): ... ValueError: the tree decomposition of a disconnected graph is not defined
Chordal graphs have treelength 1:
sage: G = graphs.RandomChordalGraph(30) sage: while not G.is_connected(): ....: G = graphs.RandomChordalGraph(30) sage: G.treelength() 1
>>> from sage.all import * >>> G = graphs.RandomChordalGraph(Integer(30)) >>> while not G.is_connected(): ... G = graphs.RandomChordalGraph(Integer(30)) >>> G.treelength() 1
Cycles have treelength \(\lceil n/3 \rceil\):
sage: [graphs.CycleGraph(n).treelength() for n in range(3, 11)] [1, 2, 2, 2, 3, 3, 3, 4]
>>> from sage.all import * >>> [graphs.CycleGraph(n).treelength() for n in range(Integer(3), Integer(11))] [1, 2, 2, 2, 3, 3, 3, 4]
- sage.graphs.graph_decompositions.tree_decomposition.treelength_lowerbound(G)[source]¶
Return a lower bound on the treelength of \(G\).
See [DG2006] for more details.
INPUT:
G– a sage Graph
EXAMPLES:
sage: from sage.graphs.graph_decompositions.tree_decomposition import treelength_lowerbound sage: G = graphs.PetersenGraph() sage: treelength_lowerbound(G) 1 sage: G.treelength() 2 sage: G = graphs.CycleGraph(5) sage: treelength_lowerbound(G) 2 sage: G.treelength() 2
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import treelength_lowerbound >>> G = graphs.PetersenGraph() >>> treelength_lowerbound(G) 1 >>> G.treelength() 2 >>> G = graphs.CycleGraph(Integer(5)) >>> treelength_lowerbound(G) 2 >>> G.treelength() 2
- sage.graphs.graph_decompositions.tree_decomposition.treewidth(g, k=None, kmin=None, certificate=False, algorithm=None, nice=False)[source]¶
Compute the treewidth of \(g\) (and provide a decomposition).
INPUT:
g– a Sage Graphk– integer (default:None); indicates the width to be considered. Whenkis an integer, the method checks that the graph has treewidth \(\leq k\). IfkisNone(default), the method computes the optimal treewidth.kmin– integer (default:None); when specified, search for a tree-decomposition of width at leastkmin. This parameter is useful when the graph can be decomposed into atoms. This parameter is ignored whenkis notNoneor whenalgorithm == 'tdlib'.certificate– boolean (default:False); whether to return the tree-decomposition itselfalgorithm– whether to use'sage'or'tdlib'(requires the installation of the sagemath_tdlib: Tree decompositions with tdlib package). The default behaviour is to use ‘tdlib’ if it is available, and Sage’s own algorithm when it is not.nice– boolean (default:False); whether or not to return the nice tree decomposition, providedcertificateisTrue
OUTPUT:
g.treewidth()returns treewidth of the graphg.When
kis specified, it returnsFalseif there is no tree decomposition of width \(\leq k\), andTrueotherwise.When
certificate=True, the tree decomposition is returned.When
nice=True, the nice tree decomposition is returned.ALGORITHM:
This function virtually explores the graph of all pairs
(vertex_cut, cc), wherevertex_cutis a vertex cut of the graph of cardinality \(\leq k+1\), andconnected_componentis a connected component of the graph induced byG-vertex_cut.We deduce that the pair
(vertex_cut, cc)is feasible with treewidth \(k\) ifccis empty, or if a vertexvfromvertex_cutcan be replaced with a vertex fromcc, such that the pair(vertex_cut+v,cc-v)is feasible.Note
The implementation would be much faster if
cc, the argument of the recursive function, was a bitset. It would also be very nice to not copy the graph in order to compute connected components, for this is really a waste of time.See also
path_decomposition()computes the pathwidth of a graph. See also thevertex_separationmodule.EXAMPLES:
The PetersenGraph has treewidth 4:
sage: petersen = graphs.PetersenGraph() sage: petersen.treewidth(algorithm='sage') 4 sage: petersen.treewidth(algorithm='sage', certificate=True) Tree decomposition: Graph on 6 vertices
>>> from sage.all import * >>> petersen = graphs.PetersenGraph() >>> petersen.treewidth(algorithm='sage') 4 >>> petersen.treewidth(algorithm='sage', certificate=True) Tree decomposition: Graph on 6 vertices
The PetersenGraph has treewidth 4 (with
tdlib):sage: petersen = graphs.PetersenGraph() sage: petersen.treewidth(algorithm='tdlib') # optional - tdlib 4 sage: petersen.treewidth(algorithm='tdlib', certificate=True) # optional - tdlib Tree decomposition: Graph on 6 vertices
>>> from sage.all import * >>> petersen = graphs.PetersenGraph() >>> petersen.treewidth(algorithm='tdlib') # optional - tdlib 4 >>> petersen.treewidth(algorithm='tdlib', certificate=True) # optional - tdlib Tree decomposition: Graph on 6 vertices
Nice tree decomposition of the PetersenGraph has 28 nodes:
sage: petersen = graphs.PetersenGraph() sage: petersen.treewidth(algorithm='sage', certificate=True, nice=True) Nice tree decomposition of Tree decomposition: Graph on 28 vertices sage: petersen.treewidth(algorithm='tdlib', certificate=True, nice=True) # optional - tdlib Nice tree decomposition of Tree decomposition: Graph on 28 vertices
>>> from sage.all import * >>> petersen = graphs.PetersenGraph() >>> petersen.treewidth(algorithm='sage', certificate=True, nice=True) Nice tree decomposition of Tree decomposition: Graph on 28 vertices >>> petersen.treewidth(algorithm='tdlib', certificate=True, nice=True) # optional - tdlib Nice tree decomposition of Tree decomposition: Graph on 28 vertices
The treewidth of a 2-dimensional grid is its smallest side:
sage: graphs.Grid2dGraph(2,5).treewidth() 2 sage: graphs.Grid2dGraph(3,5).treewidth() 3
>>> from sage.all import * >>> graphs.Grid2dGraph(Integer(2),Integer(5)).treewidth() 2 >>> graphs.Grid2dGraph(Integer(3),Integer(5)).treewidth() 3
When parameter
kminis specified, the method searches for a tree-decomposition of width at leastkmin:sage: g = graphs.PetersenGraph() sage: g.treewidth() 4 sage: g.treewidth(kmin=2, algorithm='sage') 4 sage: g.treewidth(kmin=g.order(), certificate=True, algorithm='sage') Tree decomposition: Graph on 1 vertex
>>> from sage.all import * >>> g = graphs.PetersenGraph() >>> g.treewidth() 4 >>> g.treewidth(kmin=Integer(2), algorithm='sage') 4 >>> g.treewidth(kmin=g.order(), certificate=True, algorithm='sage') Tree decomposition: Graph on 1 vertex
- sage.graphs.graph_decompositions.tree_decomposition.width_of_tree_decomposition(G, T, check=True)[source]¶
Return the width of the tree decomposition \(T\) of \(G\).
The width of a tree-decomposition is the size of the largest bag minus 1. The empty graph and a graph of order 1 have treewidth 0.
INPUT:
G– a sage GraphT– a tree-decomposition for \(G\)check– boolean (default:True); whether to check that the tree-decomposition \(T\) is valid for \(G\)
EXAMPLES:
sage: from sage.graphs.graph_decompositions.tree_decomposition import width_of_tree_decomposition sage: G = graphs.PathGraph(3) sage: T = G.treewidth(certificate=True) sage: width_of_tree_decomposition(G, T, check=True) 1
>>> from sage.all import * >>> from sage.graphs.graph_decompositions.tree_decomposition import width_of_tree_decomposition >>> G = graphs.PathGraph(Integer(3)) >>> T = G.treewidth(certificate=True) >>> width_of_tree_decomposition(G, T, check=True) 1