Shellings from Relative Shellings, with an Application to NP-Completeness

Shellings of simplicial complexes have long been a useful tool in topological and algebraic combinatorics. complex expose large amount information helpful way, but are not easy to construct, often requiring deep about the structure complex. It is natural ask whether shellings may be efficiently found computationally. In recent paper, Goaoc, Pat\'ak, Pat\'akov\'a, Tancer Wagner gave negative answer this question (assuming P \neq NP), showing that problem deciding shellable NP-complete. we give simplified constructions various gadgets used NP-completeness proof these authors. Using combined with relative shellability other ideas, also exhibit simpler decision problem. Our method systematically uses build up desired properties.