Isomorph-Free Exhaustive Generation

There are 1,132,835,421,602,062,347 nonisomorphic one‐factorizations of K14

“…We reject isomorphs among the visited one-factorizations using the framework in [7], which is an instantiation of the canonical augmentation technique developed by McKay [13]. In essence, we identify a canonical Aut(X )-orbit of seeds contained by a visited one-factorization X , and then check whether the seed (Π, T, S) from which X was extended is in the canonical orbit.…”

Listing Triconnected Rooted Plane Graphs

“…The problem of enumerating (i.e., listing) all graphs in particular classes of graphs is one of the most fundamental and important issues in graph theory, and has been studied extensively [1,7,9,10,13,16]. Cataloguing graphs, i.e., making the complete of graphs in a particular class can be used in a various way: search for a possible counterexample to a mathematical conjecture; choosing the best graph among all candidate graphs; and experiment for measuring the average performance of a graph algorithm over all possible input graphs.…”

Network Motif Discovery Using Subgraph Enumeration and Symmetry-Breaking

“…To avoid the increased complexity of subgraph enumeration (in the absence of an appropriate hashing scheme) our algorithm works by exhaustively searching for the instances of a single query graph in a network. (To find all motifs of a given size we couple this search with subgraph enumeration, using McKay's geng and directg tools [15]). Even though the subgraph isomorphism problem -finding a given graph as a subgraph of a larger network -is known to be NP-complete, several algorithmic improvements enable this search to be carried out effectively in practice, even for subgraphs up to 31 nodes (and potentially even more).…”

“…Our method can be used to find all instances of subgraphs of a given size, similar to previous exhaustive methods. To do this, we generate all non-isomorphic graphs of a particular size using McKay's geng and directg tools [15]. Then for each graph, we evaluate its significance as above.…”

“…McKay's geng and directg tools [15] were used to enumerate all graphs of a given size. Much of the computational work was carried out on the compute cluster at the Broad Institute of M.I.T.…”