Biconvex graphs: ordering and algorithms

Abstract

In this paper we show that vertices of biconvex graphs have an ordering that we call a biconvex straight ordering. The new suggested ordering has the following properties: it preserves the biconvex property, and it generalizes the strong ordering achievable for bipartite permutation graphs. Moreover, we show that such an ordering may be obtained efficiently in parallel. Additionally, we use the new ordering to solve the vertex ranking problem on biconvex graphs, and to observe that biconvex graphs are 4-polygon graphs. In a related context this ordering may be viewed as one for rows and columns of 0–1 matrices. The matrix interpretation may be stated as follows. For every 0–1 matrix that has the consecutive 1's property for both the rows and the columns, the rows and columns may be permuted so that the following is true: the matrix has the consecutive 1's property for both the rows and the columns, and does not contain the following submatrix: 0 1 1 0 .