A Note on the Stochastic Rank of a Bipartite Graph

Dulmage, A.L. ; Mendelsohn, N.S.

Canadian mathematical bulletin, Tome 2 (1959) no. 3, pp. 159-162

Résumé

A bipartite graph is a system consisting of two sets of vertices S and T and a set of edges K, each edge joining a vertex of S to a vertex of T. A set U of edges of K is said to be independent if no two edges of U have a vertex in common. The largest possible number of independent edges has been variously called the exterior dimension [3], term rank [4, 5, 7], etc. This number is the same as the smallest number of vertices in a set W such that each edge of K has at least one of its vertices in W. The edges of a finite bipartite graph can be represented as a set of cells in a matrix as follows. If S = a1, a2, ..., an T = b1, b2, ... bm, the edges of K are represented by some of the cells of an n by m matrix as follows: if K contains the edge joining ai to bj then the (i, j)th cell of the matrix represents this edge. It is convenient sometimes to represent the set K by a matrix A with real entries aij where aij = 0 if ai is not joined to bj in K and aij > 0 if ai is joined to bj in K. Any non-null graph K will have infinitely many matrix representations.

Détail

DOI : 10.4153/CMB-1959-020-4