A Novel Algorithm for Isomorphic Graph

2016

To solve the problem of establishing isomorphism two connected undirected graphs, we use the methodology of positioning graph vertices relative to each other. 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 a neighborhood k-th (0 ≤ k ≤ n − 1) level, where n is the number of graph vertices. Next, we construct an auxiliary directed graph, generated by the selected vertex. The vertices of the digraph are positionable by special characteristics — vectors, which locate each vertex of the digraph relative the found neighborhoods. This enabled to develop an algorithm, the running time of which is equal to O(n4). MCS2000: 05C85, 68Q17.

Transactions on Networks and Communications, 2017

To determine that two given undirected graphs are isomorphic, we construct for them auxiliary graphs, using the breadth-first search. This makes capability to position vertices in each digraph with respect to each other. If the given graphs are isomorphic, in each of them we can find such positionally equivalent auxiliary digraphs that have the same mutual positioning of vertices. Obviously, if the given graphs are isomorphic, then such equivalent digraphs exist. Proceeding from the arrangement of vertices in one of the digraphs, we try to determine the corresponding vertices in another digraph. As a result we develop the algorithm for constructing a bijective mapping between vertices of the given graphs if they are isomorphic. The running time of the algorithm equal to O(n 5), where n is the number of graph vertices.

Discrete Mathematics & Theoretical Computer Science, 2014

The graph isomorphism (GI) problem asks whether two given graphs are isomorphic or not. The GI problem is quite basic and simple, however, it's time complexity is a long standing open problem. The GI problem is clearly in NP, no polynomial time algorithm is known, and the GI problem is not NP-complete unless the polynomial hierarchy collapses. In this paper, we survey the computational complexity of the problem on some graph classes that have geometric characterizations. Sometimes the GI problem becomes polynomial time solvable when we add some restrictions on some graph classes. The properties of these graph classes on the boundary indicate us the essence of difficulty of the GI problem. We also show that the GI problem is as hard as the problem on general graphs even for grid unit intersection graphs on a torus, that partially solves an open problem.

International Journal of Computer Applications, 2013

For over three decades the Graph Isomorphism (GI) problem has been extensively studied by many researchers in algorithms and complexity theory. To date, there is no formal proof to classify this problem to be in the class P or the class NP. In this paper, evidence had been proposed of the existing of polynomial time algorithm based on edge classification which can be used to prove that GI is rather in the class P. General Terms Computational Mathematics, Theoretical Informatics, Theoretical Computer Science, Graph Algorithms, Algorithms. Keywords Edge Classification, Graph Isomorphism, Polynomial Algorithm, Graph Canonization . Nomenclature GI Graph isomorphism Gd Directed Graph DG Directed unweighted graph with no self-loops and no multiple edges SnD Square n-partite directed acyclic graph EBFS Edge Breadth First Search The isomorphism problem1. INTRODUCTION The Graph Isomorphism (GI) problem consists of deciding whether two given graphs are identical even if they look different in...

Despite the significant number of isomorphism algorithms presented in the literature, till now no efforts have been done to characterize their different performance in terms of matching time. It is not clear how the behavior of those algorithms varies as the type and the size of the graphs to be matched varies in case of real applications. In this paper we present a benchmarking activity for characterize the performance of a bunch of algorithms for exact isomorphism: to this purpose we use a database of graphs specifically developed for this task.

This research study the isomorphism problem of two simple planar graphs which they have the same number of vertices and edges, we used the permutations algorithm which generates a permutation from known permutation.

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.

viXra, 2015

In this paper we develop a novel characterization for isomorphism of graphs. The characterization is obtained in terms of ordered adjacency lists to be associated with two given labeled graphs. We show that the two given labeled graphs are isomorphic if and only if their associated ordered adjacency lists can be made identical by applying the action of suitable transpositions on any one of these lists. We discuss in brief the complexity of the algorithm for deciding isomorphism of graphs and show that it is of the order of the cube of number of the number of edges.

THE 11TH NATIONAL CONFERENCE ON MATHEMATICAL TECHNIQUES AND APPLICATIONS, 2019

The present paper is focusing on the connection between the Isomorphic groups and the Isomorphic graphs. We established a few results on the necessary and sufficient condition on the dimension of the graphs, which are isomorphic with respect to the isomorphic groups. We proved those results depending on the Isomorphic groups. We discussed the methods to find isomorphic graphs by using isomorphic groups. We derived few results on the isomorphism of graphs, by illustrating few examples.

ACM Journal of Experimental Algorithmics, 2019

We describe five new algorithms, named Vsep. Four of them are for the graph automorphism group and the fifth one is for finding an isomorphism between two graphs. All nonequivalent terminal nodes-discrete partitions of the search tree are stored. This is the main difference of the exact version with the known algorithms for graph automorphism group. A new strategy is used in the exact algorithm: if during its execution the computed stabilizer orbits and order get wrong values, then the algorithm continues from a new starting point losing some of the results determined so far. The new starting point is such that the results are correct. The proposed algorithms have been tested on well-known benchmark graphs and compared with three of the most competitive known tools. The results show that for some graph families the exact Vsep algorithm outperforms these algorithms, and vice-versa for some of the others. None of the tested algorithms outperform others in all cases. The heuristic vers...