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A graph is concave-round if its vertices can be circularly enumerated so thatthe closed neighbourhood of each vertex is an interval in the enumeration. Inthis work, we give a minimal forbidden induced subgraph characterization forthe class of concave-round graphs, solving a problem posed by Bang-Jensen,Huang, and Yeo [SIAM J Discrete Math, 13:179--193, 2000]. In addition, we showthat it is possible to find one such forbidden induced subgraph in linear timein any given graph that is not concave-round. As part of the analysis, weobtain characterizations by minimal forbidden submatrices for the circular-onesproperty for rows and for the circular-ones property for rows and columns andshow that, also for both variants of the property, one of the correspondingforbidden submatrices can be found (if present) in any given matrix in lineartime. We make some final remarks regarding connections to some classes ofcircular-arc graphs.

Characterization and linear-time detection of minimal obstructions to concave-round graphs and the circular-ones property