Apply Partition Tree to Compute Canonical Labelings of Graphs

This paper establishes a theoretical framework by defining a set of concepts useful for classifying graphs and computing the canonical labeling Cmax(G) of a given undirected graph G, which including the partition tree PartT(G), maximum partition tree MaxPT(G), centre subgraph Cen(G), standard regular sequence SRQ(G), standard maximum regular sequence SMRQ(G), and so on. The implementations of algorithms 1 to 5 show how to calculate them accordingly. The worst time complexities of algorithms 1, 2, 4, and 5 are O(n 2 ) respectively. The time complexity of Algorithm 3 is O(n). By Theorem 3, all leaf nodes of PartT(G) and MaxPT(G) are the regular subgraphs. By Theorem 4 and 5, there exists only one Cen(G) in G. Regular Partition Theorem 6 shows that there exists just one corresponding PartT(G), SRQ(G), MaxPT(G), and SMRQ(G). One can use Classification Theorem 7 to category graphs. Theorem 8 and 9 establish the link between the Cen(G) and the calculation of the first node u1 added into MaxQ(G) corresponding to the canonical labeling Cmax(G) of G. Further, it utilizes the Cen(G) to calculate the first node u1 added into MaxQ(G). The proposed methods can be extended to deal with the directed graphs and weighted graphs.

[1]  George Karypis,et al.  Finding Frequent Patterns in a Large Sparse Graph* , 2005, Data Mining and Knowledge Discovery.

[2]  Igor L. Markov,et al.  Graph Symmetry Detection and Canonical Labeling: Differences and Synergies , 2012, Turing-100.

[3]  László Babai,et al.  Canonical labeling of graphs , 1983, STOC.

[4]  Petteri Kaski,et al.  Engineering an Efficient Canonical Labeling Tool for Large and Sparse Graphs , 2007, ALENEX.

[5]  Vikraman Arvind,et al.  A Logspace Algorithm for Partial 2-Tree Canonization , 2008, CSR.

[6]  Wei Wang,et al.  Efficient mining of frequent subgraphs in the presence of isomorphism , 2003, Third IEEE International Conference on Data Mining.

[7]  Stefan Arnborg,et al.  Canonical representations of partial 2- and 3-trees , 1992, BIT.

[8]  Brendan D. McKay,et al.  Practical graph isomorphism, II , 2013, J. Symb. Comput..

[9]  Brendan D. McKay,et al.  Isomorph-Free Exhaustive Generation , 1998, J. Algorithms.

[10]  George Karypis,et al.  An efficient algorithm for discovering frequent subgraphs , 2004, IEEE Transactions on Knowledge and Data Engineering.

[11]  Jiawei Han,et al.  gSpan: graph-based substructure pattern mining , 2002, 2002 IEEE International Conference on Data Mining, 2002. Proceedings..

[12]  Brendan D. McKay,et al.  Computing automorphisms and canonical labellings of graphs , 1978 .

[13]  Antonio Fernández,et al.  Conauto-2.0: Fast Isomorphism Testing and Automorphism Group Computation , 2011, ArXiv.

[14]  László Babai,et al.  Canonical labelling of graphs in linear average time , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[15]  Petteri Kaski,et al.  Conflict Propagation and Component Recursion for Canonical Labeling , 2011, TAPAS.

[16]  Adolfo Piperno,et al.  Search Space Contraction in Canonical Labeling of Graphs (Preliminary Version) , 2008, ArXiv.

[17]  Q. Li,et al.  Some further development on the eigensystem approach for graph isomorphism detection , 2005, J. Frankl. Inst..

[18]  Sahar Asadi,et al.  Kavosh: a new algorithm for finding network motifs , 2009, BMC Bioinformatics.