# Search Space Contraction in Canonical Labeling of Graphs (Preliminary Version)

@article{Piperno2008SearchSC, title={Search Space Contraction in Canonical Labeling of Graphs (Preliminary Version)}, author={Adolfo Piperno}, journal={ArXiv}, year={2008}, volume={abs/0804.4881} }

The individualization-refinement paradigm for computing a canonical labeling and/or the automorphism group of a graph is investigated. New techniques are introduced with the aim of reducing the size of the associated search space. In particular, a new partition refinement algorithm is proposed, together with graph invariants having a global nature. Experimental results and comparisons with existing tools, such as nauty, reveal that the presented approach produces a huge contraction of the…

## 59 Citations

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## References

SHOWING 1-10 OF 28 REFERENCES

### Engineering an Efficient Canonical Labeling Tool for Large and Sparse Graphs

- Computer ScienceALENEX
- 2007

Within the general framework of backtracking algorithms based on individualization and refinement, data structures, subroutines, and pruning heuristics especially for fast handling of large and sparse graphs are developed.

### Canonical labeling of graphs

- MathematicsSTOC
- 1983

An algebraic approach to the problem of assigning canonical forms to graphs by computing canonical forms and the associated canonical labelings in polynomial time is announced.

### The complexity of McKay's canonical labeling algorithm

- Mathematics, Computer ScienceGroups and Computation
- 1995

An exponential lower bound is proved of the algorit hm for connected 3-regular graphs of color-class size 4 using Fürer’s construction for these graphs.

### Moderately Exponential Bound for Graph Isomorphism

- MathematicsFCT
- 1981

The first significant complexity result in graph isomorphism testing was the linear time canonical labeling of trees, found by Hopcroft and Tarjan [16~, and independently by V.N. Zemlyachenko [29].…

### Isomorphism of graphs of bounded valence can be tested in polynomial time

- Mathematics21st Annual Symposium on Foundations of Computer Science (sfcs 1980)
- 1980

Testing isomorphism of graphs of valence ≤ t is polynomial-time reducible to the color automorphism problem for groups with small simple sections, and some results on primitive permutation groups are used to show that the algorithm runs inPolynomial time.

### An optimal lower bound on the number of variables for graph identification

- Mathematics30th Annual Symposium on Foundations of Computer Science
- 1989

It is shown that Omega (n) variables are needed for first-order logic with counting to identify graphs on n vertices and the lower bound is optimal up to multiplication by a constant.

### Stabilization procedures and applications

- Mathematics, Computer Science
- 2001

This thesis introduces an efficient way of computing 1- and 2-stable partitions and presents a particular version of a k-dimensional stabilization algorithm used in chemistry.

### Monte-Carlo algorithms in graph isomorphism testing

- Mathematics
- 2006

Abstract. We present an O(V 4 log V ) coin flipping algorithm to test vertex-colored graphs with bounded color multiplicities for color-preserving isomorphism. We are also able to generate uniformly…

### Linear time algorithm for isomorphism of planar graphs (Preliminary Report)

- MathematicsSTOC '74
- 1974

The time bound for planar graph isomorphism is improved to O(|V|) time and the algorithm can be easily extended to partition a set of planar graphs into equivalence classes of isomorphic graphs in time linear in the total number of vertices in all graphs in the set.

### Isomorphism of graphs with bounded eigenvalue multiplicity

- MathematicsSTOC '82
- 1982

Two polynomial time algorithms are described which test isomorphism of undirected graphs whose eigenvalues have bounded multiplicity, if X and Y are graphs of eigenvalue multiplicity m.