Figure 9
An algorithm for recognizing 1-representable complexes. (A) Four intervals covering a linear segment (bottom) can be represented by a simplicial complex—the nerve of the cover (middle panel). The vertexes of the corresponding interval graph \(G({\mathcal {I}})\)—the 1D skeleton of \(\Sigma \)—(color-coded) are connected if their respective intervals overlap, \(I_i \cap I_j\implies v_i\lhd v_j\). The corresponding comparability graph, \({\tilde{G}}_{\lhd }({\mathcal {I}})\) is shown above, with the order indicated by arrows: \(v_i \lhd v_j\) iff there is an arrow leading from \(v_i\) to \(v_j\). (B) Given a simplicial complex \(\Sigma \), first check whether it is the clique complex of its 1D-skeleton \(G:=sk_1(\Sigma )\). If it is not, then \(\Sigma \) is not 1-representable; if it is, then check whether the complement graph of G is a comparability graph. If it is not, then \(\Sigma \) is not 1-representable. If it is, then check the \(\times \)-property: if it holds, then \(\Sigma \) is 1-representable, otherwise it is not.
