Efficient Non-isomorphic Graph Enumeration Algorithms for Subclasses of Perfect Graphs

TL;DR

This paper introduces efficient algorithms for enumerating non-isomorphic graphs within certain subclasses of perfect graphs using BDDs, enabling enumeration with constraints and polynomial time complexity.

Contribution

It presents novel BDD-based enumeration algorithms for five intersection graph classes, capable of handling large graph sizes and additional constraints.

Findings

Algorithms enumerate all graphs with n vertices efficiently.

Enumeration is polynomial in n for each class.

Algorithms handle constraints on clique size and number of edges.

Abstract

Intersection graphs are well-studied in the area of graph algorithms. Some intersection graph classes are known to have algorithms enumerating all unlabeled graphs by reverse search. Since these algorithms output graphs one by one and the numbers of graphs in these classes are vast, they work only for a small number of vertices. Binary decision diagrams (BDDs) are compact data structures for various types of data and useful for solving optimization and enumeration problems. This study proposes enumeration algorithms for five intersection graph classes, which admit $\mathrm{O}(n)$-bit string representations for their member graphs. Our algorithm for each class enumerates all unlabeled graphs with $n$ vertices over BDDs representing the binary strings in time polynomial in $n$. Moreover, our algorithms are extended to enumerate those with constraints on the maximum (bi)clique size and/or the number of edges.