Digraph representation by its path matrix The particular case of the bipartite graphs

Abstract

This paper studies the characterization of a directed graph by its path matrix. We show first that a directed acyclic graph can be uniquely associated with this matrix. The properties of the adjacency matrix of a directed acyclic 1-graph are studied and we show that the matrix of the elementary paths allows us to calculate the adjacency matrix in an efficient way. The results obtained in that first part are used in a second part to characterize a particular class of bipartite graphs named resource graphs. We show that the adjacency matrix of a resource graph of order n + m is redundant and that it can be replaced by a vector of dimension m and a matrix of dimension n×m when such a matrix is a square one of dimension (n + m)2. We show also that the path matrix of a resource graph can be reduced in the same way. These results are used to define new efficient algorithms which tranform a resource graph by insertion or deletion of nodes or edges. These operations can be used, for instance, to prevent deadlock in systems where it exists one unit of each type of resources.