Hi,
Consider the following description.
def is_matching_covered(self, matching=None, algorithm='Edmonds',
coNP_certificate=False,
solver=None, verbose=0, *, integrality_tolerance=0.001):
r"""
Check if the graph is matching covered.
A connected nontrivial graph wherein each edge participates in some
perfect matching is called a `matching` `covered` `graph`.
If a perfect matching of the graph is provided, for bipartite graph,
this method implements a linear time algorithm as proposed in [LM2024]_
that is based on the following theorem:
Given a connected bipartite graph `G[A, B]` with a perfect matching
`M`. Construct a directed graph `D` from `G` such that `V(D) := V(G)`
and for each edge in `G` direct the corresponding edge from `A` to `B`
in `D`, if it is in `M` or otherwise direct it from `B` to `A`. The
graph `G` is matching covered if and only if `D` is strongly connected.
For nonbipartite graph, if a perfect matching of the graph is provided,
this method implements an `\mathcal{O}(|V| \cdot |E|)` algorithm, where
`|V|` and `|E|` are the order and the size of the graph respectively.
This implementation is inspired by the `M`-`alternating` `tree` `search`
method explained in [LZ2001]_. For nonbipartite graph, the
implementation is based on the following theorem:
Given a nonbipartite graph `G` with a perfect matching `M`. The
graph `G` is matching covered if and only if for each edge `uv`
not in `M`, there exists an `M`-`alternating` odd length `uv`-path
starting and ending with edges not in `M`.
The time complexity may be dominated by the time needed to compute a
maximum matching of the graph, in case a perfect matching is not
provided. Also, note that for a disconnected or a trivial graph, a
:class:`ValueError` is returned.
There are two paragraphs:
1. Given a connected bipartite graph `G[A, B]` ... `D` is strongly
connected, and
2. Given a nonbipartite graph `G` ... with edges not in `M`.
Note that these two paragraphs capture two theorems related to the
algorithm that is implemented in that particular method. So, I thought it
will be better to put them in a literal block, as it will be easy to read
and will look good.
For more info, you may please have a look at this PR
<https://github.com/sagemath/sage/pull/38218>.
I just realized that there is this `\mathcal{O}(|V| \cdot |E|)` that is
there in the second theorem. So, in fact, it not only requires the
variables to be italicized, but also requires the math mode.
Thanks and warm regards,
Janmenjaya
On Tuesday, June 25, 2024 at 8:57:14 AM UTC+5:30 David Roe wrote:
> Could you explain more why you want to italicize something inside a
> literal block?
> David
>
> On Sat, Jun 22, 2024 at 10:39 PM Janmenjaya Panda <janmenjay...@gmail.com>
> wrote:
>
>> Could someone please mention, how to italicize a particular term inside a
>> literal block?
>> Use of `term` results as it is in the HTML document instead of
>> italicizing it.
>>
>> Thanks and warm regards,
>> Janmenjaya
>>
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