# Linear Extensions of Directed Acyclic Graphs.¶

A linear extension of a directed acyclic graph is a total (linear) ordering on the vertices that is compatible with the graph in the following sense: if there is a path from x to y in the graph, the x appears before y in the linear extension.

The algorithm implemented in this module is from “Generating Linear Extensions Fast” by Preusse and Ruskey, which can be found at http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.3057 . The algorithm generates the extensions in constant amortized time (CAT) – a constant amount of time per extension generated, or linear in the number of extensions generated.

EXAMPLES:

Here we generate the 5 linear extensions of the following directed acyclic graph:

sage: from sage.graphs.linearextensions import LinearExtensions
sage: D = DiGraph({ 0:[1,2], 1:[3], 2:[3,4] })
sage: D.is_directed_acyclic()
True
sage: LinearExtensions(D).list()
[[0, 1, 2, 3, 4],
[0, 1, 2, 4, 3],
[0, 2, 1, 3, 4],
[0, 2, 1, 4, 3],
[0, 2, 4, 1, 3]]


Notice how all of the total orders are compatible with the ordering induced from the graph.

We can also get at the linear extensions directly from the graph. From the graph, the linear extensions are known as topological sorts

sage: D.topological_sort_generator()
[[0, 1, 2, 3, 4],
[0, 1, 2, 4, 3],
[0, 2, 1, 3, 4],
[0, 2, 1, 4, 3],
[0, 2, 4, 1, 3]]

class sage.graphs.linearextensions.LinearExtensions(dag)

Creates an object representing the class of all linear extensions of the directed acyclic graph code{dag}.

Note that upon construction of this object some pre-computation is done. This is the “preprocessing routine” found in Figure 7 of “Generating Linear Extensions Fast” by Preusse and Ruskey.

This is an in-place algorithm and the list self.le keeps track of the current linear extensions. The boolean variable self.is_plus keeps track of the “sign”.

EXAMPLES:

sage: from sage.graphs.linearextensions import LinearExtensions
sage: D = DiGraph({ 0:[1,2], 1:[3], 2:[3,4] })
sage: l = LinearExtensions(D)
True

generate_linear_extensions(i)

This a Python version of the GenLE routine found in Figure 8 of “Generating Linear Extensions Fast” by Pruesse and Ruskey.

Note that this is meant to be called by the list method and is not meant to be used directly.

EXAMPLES:

sage: from sage.graphs.linearextensions import LinearExtensions
sage: D = DiGraph({ 0:[1,2], 1:[3], 2:[3,4] })
sage: l = LinearExtensions(D)
sage: l.linear_extensions = []
sage: l.linear_extensions.append(l.le[:])
sage: l.generate_linear_extensions(l.max_pair)
sage: l.linear_extensions
[[0, 1, 2, 3, 4], [0, 2, 1, 3, 4]]

incomparable(x, y)

Returns True if vertices x and y are incomparable in the directed acyclic graph when thought of as a poset.

EXAMPLES:

sage: from sage.graphs.linearextensions import LinearExtensions
sage: D = DiGraph({ 0:[1,2], 1:[3], 2:[3,4] })
sage: l = LinearExtensions(D)
sage: l.incomparable(0,1)
False
sage: l.incomparable(1,2)
True

list()

Returns a list of the linear extensions of the directed acyclic graph.

Note that once they are computed, the linear extensions are cached in this object.

EXAMPLES:

sage: from sage.graphs.linearextensions import LinearExtensions
sage: D = DiGraph({ 0:[1,2], 1:[3], 2:[3,4] })
sage: LinearExtensions(D).list()
[[0, 1, 2, 3, 4],
[0, 1, 2, 4, 3],
[0, 2, 1, 3, 4],
[0, 2, 1, 4, 3],
[0, 2, 4, 1, 3]]

move(element, direction)

This implements the Move procedure described on page 7 of “Generating Linear Extensions Fast” by Pruesse and Ruskey.

If direction is “left”, then this transposes element with the element on its left. If the direction is “right”, then this transposes element with the element on its right.

Note that this is meant to be called by the generate_linear_extensions method and is not meant to be used directly.

EXAMPLES:

sage: from sage.graphs.linearextensions import LinearExtensions
sage: D = DiGraph({ 0:[1,2], 1:[3], 2:[3,4] })
sage: l = LinearExtensions(D)
sage: _ = l.list()
sage: l.le = [0, 1, 2, 3, 4]
sage: l.move(1, "left")
sage: l.le
[1, 0, 2, 3, 4]
sage: l.move(1, "right")
sage: l.le
[0, 1, 2, 3, 4]

right(i, letter)

If letter ==”b”, then this returns True if and only if self.b[i] is incomparable with the elements to its right in self.le. If letter == “a”, then it returns True if and only if self.a[i] is incomparable with the element to its right in self.le and the element to the right is not self.b[i]

This is the Right function described on page 8 of “Generating Linear Extensions Fast” by Pruesse and Ruskey.

Note that this is meant to be called by the generate_linear_extensions method and is not meant to be used directly.

EXAMPLES:

sage: from sage.graphs.linearextensions import LinearExtensions
sage: D = DiGraph({ 0:[1,2], 1:[3], 2:[3,4] })
sage: l = LinearExtensions(D)
sage: _ = l.list()
sage: l.le
[0, 1, 2, 4, 3]
sage: l.a
[1, 4]
sage: l.b
[2, 3]
sage: l.right(0, "a")
False
sage: l.right(1, "a")
False
sage: l.right(0, "b")
False
sage: l.right(1, "b")
False

switch(i)

This implements the Switch procedure described on page 7 of “Generating Linear Extensions Fast” by Pruesse and Ruskey.

If i == -1, then the sign is changed. If i > 0, then self.a[i] and self.b[i] are transposed.

Note that this meant to be called by the generate_linear_extensions method and is not meant to be used directly.

EXAMPLES:

sage: from sage.graphs.linearextensions import LinearExtensions
sage: D = DiGraph({ 0:[1,2], 1:[3], 2:[3,4] })
sage: l = LinearExtensions(D)
sage: _ = l.list()
sage: l.le = [0, 1, 2, 3, 4]
sage: l.is_plus
True
sage: l.switch(-1)
sage: l.is_plus
False
sage: l.a
[1, 4]
sage: l.b
[2, 3]
sage: l.switch(0)
sage: l.le
[0, 2, 1, 3, 4]
sage: l.a
[2, 4]
sage: l.b
[1, 3]


Genus

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