On the complexity of recognizing Hamming graphs and related classes of graphs

This paper contains a new algorithm that recognizes whether a given graph G is a Hamming graph, i.e. a Cartesian product of complete graphs, in O(m) time and O(n2) space. Here m and n denote the numbers of edges and vertices of G, respectively. Previously this was only possible in O(m log n) time. Moreover, we present a survey of other recognition algorithms for Hamming graphs, retracts of Hamming graphs and isometric subgraphs of Hamming graphs. Special emphasis is also given to the bipartite case in which these classes are reduced to binary Hamming graphs, median graphs and partial binary Hamming graphs.