On the graph isomorphism problem

2016

We develop the methodology of positioning graph vertices relative to each other to solve the problem of determining isomorphism of two undirected graphs. Based on the position of the vertex in one of the graphs, it is determined the corresponding vertex in the other graph. For the selected vertex of the undirected graph, we define the neighborhoods of the vertices. Next, we construct the auxiliary directed graph, spawned by the selected vertex. The vertices of the digraph are positioned by special characteristics --- vectors, which locate each vertex of the digraph relative the found neighborhoods. This enabled to develop the algorithm for determining graph isomorphism, the runing time of which is equal to $O(n^4)$.

Lecture Notes in Computer Science, 2018

In this paper, we study the parallel and the space complexity of the graph isomorphism problem (GI) for several parameterizations. Let H = {H1, H2, • • • , H l } be a finite set of graphs where |V (Hi)| ≤ d for all i and for some constant d. Let G be an H-free graph class i.e., none of the graphs G ∈ G contain any H ∈ H as an induced subgraph. We show that GI parameterized by vertex deletion distance to G is in a parameterized version of AC 1 , denoted Para-AC 1 , provided the colored graph isomorphism problem for graphs in G is in AC 1. From this, we deduce that GI parameterized by the vertex deletion distance to cographs is in Para-AC 1. The parallel parameterized complexity of GI parameterized by the size of a feedback vertex set remains an open problem. Towards this direction we show that the graph isomorphism problem is in Para-TC 0 when parameterized by vertex cover or by twin-cover. Let G ′ be a graph class such that recognizing graphs from G ′ and the colored version of GI for G ′ is in logspace (L). We show that GI for bounded vertex deletion distance to G ′ is in L. From this, we obtain logspace algorithms for GI for graphs with bounded vertex deletion distance to interval graphs and graphs with bounded vertex deletion distance to cographs.

2005

In this paper we obtain two necessary-sufficient conditions for the isomorphism of graphs and propose two polynomial time algorithms based on them. We associate the so called bitableaux with the graphs, which are the usual adjacency and incidence matrices used in the matrix representation of graphs expressed in a bi-tabular form. We achieve, by application of suitable permutations (transpositions), the so called standard representation for these bitableaux to characterize the graphs uniquely. We also propose a polynomial time algorithm for the k-clique problem towards the end of the paper.

17th International Conference on Electronics, Communications and Computers (CONIELECOMP'07), 2007

Subgraph isomorphism (SI) detection is an important problem for several computer science subfields. In this paper we present a study of the Subgraph Isomorphism Problem (SIP) and its relation with the Hamiltonian cycles and SAT problems. In particular, we describe how instances of those problems can be solved throughout SI detection (using problems reductions). In our experiments we use an algorithm developed by the authors, which is capable to find all valid mappings in a SI instance. We performed several experiments, including cases for which there exists a known solution in polynomial time. In our analysis, we show the advantage and disadvantage of using a SI representation to solve Hamiltonian cycles and SAT problems.

We develop the methodology of positioning graph vertices relative to each other to solve the problem of determining isomorphism of two undirected graphs. Based on the position of the vertex in one of the graphs, it is determined the corresponding vertex in the other graph. For the selected vertex of the undirected graph, we define the neighborhoods of the vertices. Next, we construct the auxiliary directed graph, spawned by the selected vertex. The vertices of the digraph are positioned by special characteristics --- vectors, which locate each vertex of the digraph relative the found neighborhoods. This enabled to develop the algorithm for determining graph isomorphism, the runing time of which is equal to $O(n^4)$.

Discrete Applied Mathematics, 2014

In this paper we study the problem of deciding the existence of a subgraph epimorphism between two graphs. Our interest in this variant of graph matching problem stems from the study of model reductions in systems biology, where large systems of biochemical reactions can be naturally represented by bipartite digraphs of species and reactions. In this setting, model reduction can be formalized as the existence of a sequence of vertex deletion and merge operations that transforms a first reaction graph into a second graph. This problem is in turn equivalent to the existence of a subgraph (corresponding to delete operations) epimorphism (i.e. surjective homomorphism, corresponding to merge operations) from the first graph to the second. In this paper, we study theoretical properties of subgraph epimorphisms in general directed graphs. We first characterize subgraph epimorphisms (SEPI), subgraph isomorphisms (SISO) and graph epimorphisms (EPI) in terms of graph transformation operations. Then we study the graph distance measures induced by these transformations. We show that they define metrics on graphs and compare them. On the algorithmic side, we show that the SEPI existence problem is NP-complete by reduction of SAT, and present a constraint satisfaction algorithm that has been successfully used to solve practical SEPI problems on a large benchmark of reaction graphs from systems biology.

Language and Automata Theory and Applications, 2014

It is well known that problems encoded with circuits or formulas generally gain an exponential complexity blow-up compared to their original complexity. We introduce a new way for encoding graph problems, based on CNF or DNF formulas. We show that contrary to the other existing succinct models, there are examples of problems whose complexity does not increase when encoded in the new form, or increases to an intermediate complexity class less powerful than the exponential blow up. We also study the complexity of the succinct versions of the Graph Isomorphism problem. We show that all the versions are hard for PSPACE. Although the exact complexity of these problems is still unknown, we show that under most existing succinct models the different versions of the problem are equivalent. We also give an algorithm for the DNF encoded version of GI whose running time depends only on the size of the succinct representation.

Electronic Notes in Discrete Mathematics, 2005

Two vertices of a graph are said to be similar if there exists a graph automorphism mapping one of them into the other. Procedures aiming to separate vertices of a graph into equivalence classes accordingly to their similarities are the basis of many isomorphism-testing algorithms. In practice, these procedures efficiently reduce the search space of backtracking-based techniques. We present a new refinement procedure that associates to each pair of vertices the value of an arbitrary predefined invariant. We also propose such a new invariant that we call the least colored shortest path.

Given two graphs $G_1$ and $G_2$ on $n$ vertices each, we define a graph $G$ on vertex set $V_1\times V_2$ and the edge set as the union of edges of $G_1\times \bar{G_2}$, $\bar{G_1}\times G_2$, $\{(v,u'),(v,u"))(|u',u"\in V_2\}$ for each $v\in V_1$, and $\{((u',v),(u",v))|u',u"\in V_1\}$ for each $v\in V_2$. We consider the completely-positive Lov\'asz $\vartheta$ function, i.e., $cp\vartheta$ function for $G$. We show that the function evaluates to $n$ whenever $G_1$ and $G_2$ are isomorphic and to less than $n-1/(4n^4)$ when non-isomorphic. Hence this function provides a test for graph isomorphism. We also provide some geometric insight into the feasible region of the completely positive program.

2010

The Graph Isomorphism problem restricted to graphs of bounded treewidth or bounded tree distance width are known to be solvable in polynomial time [2],[19]. We give restricted space algorithms for these problems proving the following results: • Isomorphism for bounded tree distance width graphs is in L and thus complete for the class. We also show that for this kind of graphs a canon can be computed within logspace. • For bounded treewidth graphs, when both input graphs are given together with a tree decomposition, the problem of whether there is an isomorphism which respects the decompositions (i.e. considering only isomorphisms mapping bags in one decomposition blockwise onto bags in the other decomposition) is in L. • For bounded treewidth graphs, when one of the input graphs is given with a tree decomposition the isomorphism problem is in LogCFL. • As a corollary the isomorphism problem for bounded treewidth graphs is in LogCFL. This improves the known TC 1 upper bound for the problem given by Grohe and Verbitsky [8].