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

SAFE, MARTÍN D.

JOURNAL OF GRAPH THEORY

JOHN WILEY & SONS INC

Año: 2020 vol. 93 p. 268 - 298

0364-9024

A graph is concave‐round if its vertices can be circularly enumerated so that the closed neighborhood of each vertex is an interval in the enumeration. In this study, we give a minimal forbidden induced subgraph characterization for the class of concave‐round graphs, solving a problem posed by Bang‐Jensen, Huang, and Yeo [SIAM J. Discrete Math., 13 (2000), pp. 179?193]. In addition, we show that it is possible to find one such forbidden induced subgraph in linear time in any givengraph that is not concave‐round. As part of the analysis, we obtain characterizations by minimal forbidden submatrices for the circular‐ones property for rows and for the circular‐ones property for rows and columns and show that, also for both variants of the property, oneof the corresponding forbidden submatrices can be found (if present) in any given matrix in linear time. We make some final remarks regarding connections to some classes of circular‐arc graphs.