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

Abstract

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 given graph 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, one of 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.

Counting Kernels in Directed Graphs with Arbitrary Orientations

“…Concave-round graphs are quasi-line. They form a subclass of normal circular-arc graphs [Saf20] and a proper superclass of CIGs. Unknowingly, we have already solved the problem for this class, because of the following easy but unpublished inclusion.…”

“…Proof. A concave-round graph is (at least) one of circular interval and co-bipartite [Tuc71;Saf20]. In turn, co-bipartite graphs (i.e.…”

A Heuristic Approach to the Consecutive Ones Submatrix Problem

“…C1P has been extensively investigated. Kendall 7,8 indicated that the first study of the property was introduced by an archaeologist "Flinders Petrie" in the 19th century. Some heuristic approaches were established for this problem before the first polynomial complexity solution that was proposed by Fulkerson and Gross 1 .…”