Parallel Recognition and Location Algorithms for Chordal Graphs Using Distance Matrices

We present efficient parallel algorithms for recognizing chordal graphs and locating all maximal cliques of a chordal graph G=(V,E). Our techniques are based on partitioning the vertex set V using information contained in the distance matrix of the graph. We use these properties to formulate parallel algorithms which, given a graph G=(V,E) and its adjacency-level sets, decide whether or not G is a chordal graph, and, if so, locate all maximal cliques of the graph in time O(k) by using 82,n2/k processors on a CRCW-PRAM, where 8 is the maximum degree of a vertex in G and 1 <k_<n. The construction of the adjacency-level sets can be done by computing first the distance matrix of the graph, in time O(logn) with O(n~D~) processors, where D e is the output size of the partitions and 13=2.376, and then extracting all necessary set information. Hence, the overall time and processor complexity of both algorithms are O(logn) and O(max{62,n 2/logn, nl3+Dc} ), respectively. These results imply that, for 6<4nlogn, the proposed algorithms improve in performance upon the best-known algorithms for these problems.