Max-tolerance graphs as intersection graphs: cliques, cycles, and recognition

Abstract

Max-tolerance graphs can be regarded as generalized interval graphs, where two intervals Ii and Ij only induce an edge in the corresponding graph iff they overlap for an amount of at least max{ti, tj} where ti is an individual tolerance parameter associated to each interval Ii. A new geometric characterization of max-tolerance graphs as intersection graphs of isosceles right triangles, shortly called semi-squares, leverages the solution of various graph-theoretic problems in connection with max-tolerance graphs. First, we solve the maximal and maximum cliques problem. It arises naturally in DNA sequence analysis where the maximal cliques might be interpreted as functional domains carrying biologically meaningful information. We prove an upper bound of O(n3) for the number of maximal cliques in max-tolerance graphs and give an efficient O(n3) algorithm for their computation. In the same vein, the semi-square representation yields a simple proof for the fact that this bound is asymptotically tight, i.e., a class of max-tolerance graphs is presented where the instances have Ω(n3) maximal cliques. Additionally, we answer an open question posed in [8] by showing that max-tolerance graphs do not contain complements of cycles Cn for n > 9. By exploiting the new representation more deeply, we can go even further and prove that the recognition problem for max-tolerance graphs is NP-hard.