Topological combinatorics in the view of substructures and minimal counterexamples of simplicial complexes

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This study aimed analyses of combinatorial properties of simplicial complexes based on the structural analysis of substructures and minimal counterexamples. Especially, we studied shellability, sequential Cohen-Macaulayness, and partitionability, which play important roles in the study of topological combinatorics, and showed the counterexamples for these properties with respect to the restriction of the vertex set coincide for the class of simplicial complexes of dimension at most two, or in the class of flag complexes. We also studied the pure-skeletons of these counterexamples, and also proceeded some related applicational studies.