In Theorem 5 of [1] it is proved that for any graph G if S a convex subset of vertices of G, then the convex hull of the contour set of S equals the original set S: S = co(Ct(S)).
As noted by the authors this a similar, more general, property to the classical Minkowski-Krein-Milman property defined in terms of extreme vertices.
Does this theorem hold for hypergraphs as well?
I think so, but I haven't yet found any references. I would appreciate if somone could clarify this.
Thank you.
Kindest regards,
Marcos.
[1] Cáceres, J., Márquez, A., Oellerman, O.R., Puertas, M.L.: Rebuilding convex sets in graphs. Discrete Mathematics 297(1), 26-37 (2005), Elsevier. https://doi.org/10.1016/j.disc.2005.03.020
In [1] following statement is made:
"In graphs with the Minkowski–Krein–Milman property, the set of extreme vertices for a convex set S is minimal in the sense that any extreme point of S is not in the convex hull of a subset of S that does not contain it. Unfortunately the contour set does not share this property in general."
Following link might be useful:
https://arxiv.org/pdf/1407.3744.pdf
Thanks Waqas Jamil for your answer.
What I interpret from the statement: "Unfortunately the countour set does not share this property in general", is that for *graphs* in general the countour set is not minimal as the extreme set.
But I am not sure still if *in general* also includes hypergraphs, or just graphs. By your answer, I infer that you think the statement applies to hypergraphs as well. Right?
Thanks
Kindest regards,
Marcos
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