Listing Chordal Graphs and Interval Graphs

Abstract

We propose three algorithms for enumeration problems; given a graph G, to find every chordal supergraph (in K _n ) of G, to find every interval supergraph (in K _n ) of G, and to find every interval subgraph of G in K _n . The algorithms are based on the reverse search method. A graph is chordal if and only if it has no induced chordless cycle of length more than three. A graph is an interval graph if and only if it has an interval representation. To the best of our knowledge, ours are the first results about the enumeration problems to list every interval subgraph of the input graph and to list every chordal/interval supergraph of the input graph in polynomial time. The time complexities of the first algorithm is O((n+m)²) for each output graph, and those for the rest two algorithms are O(n ³) for each output graph, where m is the number of edges of input graph G. We also show that a straight-forward depth-first search type algorithm is not appropriate for these problems.