Embeddability and Stresses of Graphs
نویسنده
چکیده
Gluck [11] has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that linklessly embeddable graphs are generically 4-stress free. Both of these results are corollaries of the following theorem: every Kr+2-minor free graph is generically r-stress free for 1 ≤ r ≤ 4. (This assertion is false for r ≥ 6.) We give an equivalent formulation of this theorem in the language of symmetric algebraic shifting and show that its analogue for exterior algebraic shifting also holds. Some further extensions are detailed.
منابع مشابه
On embeddability and stresses of graphs
Gluck [6] has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that already the K5-minor freeness guarantees the stress freeness. More generally, we prove that every Kr+2-minor free graph is generically r-stress free for 1 ≤ r ≤ 4. (This assertion is false for r ≥ 6.) Some further extensions are discussed.
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