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On a famous website I've seen the following: The skeletons of the simple polyhedra correspond to the triangulated graphs, the smallest of which are illustrated above. That "illustration above" shows only triangulated planar graphs.

  • Does it mean that the skeleton of a square pyramide or a triangular prism is a also a triangulation?

or does "correspond" mean:

  • a triangulated graph is the skeleton of a simple polyhedron?

or is it a wrong formulation?

@Manfred Weis: What confuses me is the last sentence on the Simple polyhedron page on Wolfram: "The skeletons of the simple polyhedra correspond to the triangulated graphs,..." https://mathworld.wolfram.com/SimplePolyhedron.html

I understand the contrary: " triangulated graphs are the skeletons of simple polyhedra" but not the formulation above in Italics.

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    $\begingroup$ At least on Wolfram the class of graphs that are defined to be triangulated are the planar graphs: "A planar graph G is said to be triangulated (also called maximal planar) if the addition of any edge to G results in a nonplanar graph. If the special cases of the triangle graph C_3 and tetrahedral graph K_4 (which are planar that already contain a maximal number of edges) are included, maximal planar graphs are the skeletons of simple polyhedra and are isomorphic to planar graphs with 3n-6 edges." $\endgroup$
    – Manfred Weis
    Commented Mar 3, 2023 at 15:03
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    $\begingroup$ I think Wolfram is conflating simple polyhedra and simplicial polyhedra. A simplicial polyhedron has only triangular faces. See en.wikipedia.org/wiki/Simplicial_polytope $\endgroup$
    – Gerry Myerson
    Commented Mar 5, 2023 at 22:17
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    $\begingroup$ Yes, Wolfram is wrong. The skeleton of a simple polyhedron is a 3-connected planar graph with all vertices of degree 3, whereas the skeleton of a simplicial polyhedron is a 3-connected planar graph with all faces of size 3. These are dual properties so the relation is close not not equality. $\endgroup$
    – Brendan McKay
    Commented Mar 6, 2023 at 0:36

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