Every shellable d-dimensional simplicial complex with at most d+3 vertices is extendably shellable, proved through exposed-edge deletions in chordal graphs.
Non-ridge-chordal complexes whose clique complex has shellable Alexander dual
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abstract
A recent conjecture that appeared in three papers by Bigdeli--Faridi, Dochtermann, and Nikseresht, is that every simplicial complex whose clique complex has shellable Alexander dual, is ridge-chordal. This strengthens the long-standing Simon's conjecture that the $k$-skeleton of the simplex is extendably shellable, for any $k$. We show that the stronger conjecture has a negative answer, by exhibiting an infinite family of counterexamples.
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Extendable shellability for $d$-dimensional complexes on $d+3$ vertices
Every shellable d-dimensional simplicial complex with at most d+3 vertices is extendably shellable, proved through exposed-edge deletions in chordal graphs.