An Efficient PQ-Graph Algorithm for Solving the Graph-Realization Problem

Abstract A (0, 1)-matrix is called graphic if it is a fundamental circuit matrix of a graph. Given a (0, 1)-matrix N , the graph-realization problem is (i) to determine whether N is graphic and (ii) if graphic, to realize a graph which has N as its fundamental circuit matrix. We propose a data structure called a PQ -graph based on PQ -trees and then present an efficient algorithm for solving the graph-realization problem by means of PQ -graphs. A running time required for the algorithm is O (νvα(ν, k )), where v is the number of nonzero elements of a given (0, 1)-matrix N , k is the number of rows of N and α(·, ·) is a function defined in terms of Akermann's function. Since the value of α(ν, k ) is not more than 3 for all practical values of ν and k , we can solve the graph-realization problem in a running time almost proportional to ν, the number of nonzero elements of N .