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February 5

Identifying a vertex that is in every maximum matching of a bipartite graph

A bipartite graph is never factor-critical, and so must have a vertex v that is in every maximum matching. Is there any characterisation of such vertices/way to identify them? --2404:2000:2000:5:0:0:0:C2 (talk) 02:02, 5 February 2015 (UTC)[reply]

Actually simply identifying whether one of the partitions contains such a vertex would be sufficient, if that's possible. — Preceding unsigned comment added by 2404:2000:2000:5:0:0:0:C2 (talk) 02:03, 5 February 2015 (UTC)[reply]

Name of the shape in between three circles?

Is there a special name for the triangular shaped area that is made from three touching circles of the same size? --HappyCamper 09:16, 5 February 2015 (UTC)[reply]

Reuleaux triangle? ---Sluzzelin talk 10:28, 5 February 2015 (UTC)[reply]

I suppose @HappyCamper: asks for a name of the green figure on this image

https://www.illustrativemathematics.org/illustrations/1006

but I do not know its name (if any). --CiaPan (talk) 12:28, 5 February 2015 (UTC)[reply]

Thanks @CiaPan: for the clarification, this was exactly what I was thinking. --HappyCamper 20:56, 6 February 2015 (UTC)[reply]

Yep, sorry, hadn't read carefully. ---Sluzzelin talk 13:50, 5 February 2015 (UTC)[reply]

I don't know a name for the figure itself, but it generates an Infinite-order triangular tiling of the hyperbolic disc, and there are many nice pictures there. Sławomir Biały (talk) 12:41, 5 February 2015 (UTC)[reply]

That's a tiling by (ideal) triangles, with straight sides. The Poincaré disc projection just happens to distort straight lines into curves. Even in that projection, at most one of the projected triangles will have the shape specified in the original question, and in the Klein projection or upper-half-plane projection of the same tiling, none will. So I think it's really misleading to say that this shape tiles the hyperbolic plane. -- BenRG (talk) 20:58, 5 February 2015 (UTC)[reply]

I don't understand the nature of your objection. Do you disagree that the region of interest is equivalent, under the hyperbolic group acting on the unit disc, to a fundamental domain of that tiling? Sławomir Biały (talk) 22:24, 5 February 2015 (UTC)[reply]

A similar, though AFAICT different, curve is the Deltoid, aka a Hypocycloid of 3 cusps. There may be useful information in these articles. -- Meni Rosenfeld (talk) 18:33, 5 February 2015 (UTC)[reply]

In hyperbolic geometry, an equilateral ideal triangle can assume this shape. Informally, I have seen it called a golf tee. The shape plays a role in the construction of the Apollonian gasket, but I haven't seen the shape named there. --Mark viking (talk) 19:56, 5 February 2015 (UTC)[reply]

See my reply to Sławomir Biały above—that's an artifact of the Poincaré disk projection, not an actual shape assumed by triangles in the hyperbolic plane. -- BenRG (talk) 20:58, 5 February 2015 (UTC)[reply]

It's a Pseudotriangle. For instance, Fig. 1.1 of this paper http://www.yuugen.jp/doc/thesis.pdf shows the pseudotriangle created by three touching circles. --Modocc (talk) 22:37, 5 February 2015 (UTC)[reply]

But it's only one of many pseudotriangles. The deltoid is also a pseudotriangle. I don't know a name for the figure we're talking about either. Perhaps if HappyCamper can produce an original mathematical theorem about the shape, it could be named HappyCamper's Curve. --65.94.50.4 (talk) 22:48, 5 February 2015 (UTC)[reply]

True, but at least it has a name. --Modocc (talk) 22:54, 5 February 2015 (UTC)[reply]

Pseudotriangle....Hmm, I've never heard of this before, but I guess it will do. Thanks! --HappyCamper 20:56, 6 February 2015 (UTC)[reply]