Constructing Hard Examples for Graph Isomorphism

Computing Roots of Directed Graphs is Graph Isomorphism Hard

The k-th power D of a directed graph D is defined to be the directed graph on the vertices of D with an arc from a to b in D iff one can get from a to b in D with exactly k steps. This notion is equivalent to the k-fold composition of binary relations or k-th powers of Boolean matrices. A k-th root of a directed graph D is another directed graph R with R = D. We show that for each k ≥ 2, comput...

Accelerations for Graph Isomorphism

In this paper, we present two main results. First, by only one conjecture (Conjecture 2.9) for recognizing a vertex symmetric graph, which is the hardest task for our problem, we construct an algorithm for finding an isomorphism between two graphs in polynomial time O(n). Second, without that conjecture, we prove the algorithm to be of quasi-polynomial time O(n ). The conjectures in this paper ...

On Graph Isomorphism for Restricted Graph Classes

Graph isomorphism (GI) is one of the few remaining problems in NP whose complexity status couldn’t be solved by classifying it as being either NP-complete or solvable in P. Nevertheless, efficient (polynomial-time or even NC) algorithms for restricted versions of GI have been found over the last four decades. Depending on the graph class, the design and analysis of algorithms for GI use tools f...

Graph Isomorphism

While in general it is not known whether there is a polynomial time algorithm to decide whether two given graphs are isomorphic, there are polynomial-time algorithms for certain subsets of graphs, including but not limited to planar graphs and graphs with bounded valence. In this thesis, we will give a brief introduction on the Graph Isomorphism Problem and its relations to complexity theory. W...

Completeness results for graph isomorphism