On Mon, Mar 11, 2024 at 9:24 PM Samuel Lubliner <samlubli...@gmail.com>
wrote:
> I've been exploring the concept of antisymmetry in DiGraphs within
> SageMath and noticed a discrepancy between the standard mathematical
> definition of an antisymmetric relation and SageMath's implementation for
> DiGraphs. I'm looking for some clarification or insight into this
> observation.
>
> The standard definition of antisymmetry in a relation R states:
> if aRb and bRa, then a=b.
>
> In contrast, Sage seems to interpret antisymmetry for DiGraphs in a way
> that emphasizes the absence of reciprocal paths, which is more restrictive.
>
> To illustrate, I ran a few tests within a SageCell to understand how
> antisymmetric() behaves with different graph configurations:
>
> A graph with a loop and no reciprocal edges, which should be antisymmetric:
> ```
> DiGraph([(1, 2), (3, 1), (1, 1)], loops=True).antisymmetric() # Expected
> True, Returns True
> ```
>
>
> A graph with a direct reciprocal relationship (1,2) and (2,1), clearly
> violating antisymmetry:
> ```
> DiGraph([(1, 2), (2, 1), (3, 1), (1, 1)], loops=True).antisymmetric() #
> Expected False, Returns False
> ```
>
> This third example is interesting because, under the standard mathematical
> definition, antisymmetry focuses on direct reciprocal relations between
> pairs of elements, not the existence of a path between vertices. Therefore,
> a cycle does not inherently violate antisymmetry unless there are direct
> reciprocal edges between any two vertices in the graph.
> ```
> DiGraph([(1, 2), (2, 3), (3, 4), (4, 1)]).antisymmetric() # Expected True,
> Returns False
> ```
>
>
you can check the docs, and see that Sage, essentially, calls directed
acyclic graphs antisymmetric.
I.e. if there is a path from x to y then there is no path from y to x
(assuming x!=y)
A graph represents an antisymmetric relation if the existence of a
path
from a vertex `x` to a vertex `y` implies that there is not a path
from
`y` to `x` unless `x = y`.
that's a more interesting, mathematically, definition, than mere absense of
loops of length 2.
> Is SageMath's antisymmetric() method intentionally designed to consider
> the broader structure of the graph by evaluating paths rather than just
> direct edges to determine antisymmetry? It would be great to get some
> clarification on this and understand the rationale behind SageMath's
> implementation choice.
>
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