Isomorphism Testing Research Articles

Compact cellular algebras and permutation groups

Having in mind the generalization of Birkhoff's theorem on doubly stochastic matrices we define compact cellular algebras and compact permutation groups. Arising in this connection weakly compact graphs extend compact graphs introduced by G. Tinhofer. It is proved that compact algebras are exactly the centralizer algebras of compact groups. The technique developed enables us to get nontrivial examples of compact algebras and groups as well as completely identify compact Frobenius groups and the adjacency algebras of Johnson's and Hamming's schemes. In particular, Petersen's graph proves to be not compact, which answers a question by C. Godsil. Simple polynomial-time isomorphism tests for the classes of compact cellular algebras and weakly compact graphs are presented.

Isomorph-Free Exhaustive Generation

We describe a very general technique for generating families of combinatorial objects without isomorphs. It applies to almost any class of objects for which an inductive construction process exists. In one form of our technique, no explicit isomorphism testing is required. In the other form, isomorph testing is restricted to within small subsets of the entire set of objects. A variety of different examples are presented, including the generation of graphs with some hereditary property, the generation of Latin rectangles and the generation of balanced incomplete block designs. The technique can also be used to perform unbiased statistical analysis, including approximate counting, of sets of objects too large to generate exhaustively.

PSEUDO PROBABILISTIC APPROACH TO TEST ISOMORPHISM AMONG KINEMATIC CHAINS

There is no dearth of methods to test isomorphism amongst kinematic chains. Search for a computationally easier, logically simple and unique method is still on. Present work is in quest of a reliable test to detect isomorphism among kinematic chains. Work presented here is more versatile as it incorporates more features of the kinematic chain which were not included earlier such as number and type of links, their relative dispositions in the kinematic chain, nature of adjacent links etc. The method proposed is based on the concept of pseudo-probability (pseudo means it appears to be, but not exactly. The approach does not follow in-toto the principles of probability and considerable liberty has been taken in interpreting the word probability hence the word pseudo is used along with the probability schemes). Using the resemblance of different coloured balls in an urn for the number and type of links in a kinematic chain, a matrix (named P-Matrix) representing the kinematic chain in totality is generated. For the sake of comparison a numerical scheme named, pseudo probability scheme, P-Scheme, is developed from the above P-Matrix and is used for testing isomorphism. In fact the method is more powerful in the sense that each row of the proposed P-Matrix is capable of representing the respective kinematic chain distinctly and can be used to compare the kinematic chains with same link assortments, uniquely. The proposed method, besides possessing the potential of testing the isomorphism among simple-joint, single degree of freedom kinematic chains is also capable of multi degrees of freedom and multiple-joint kinematic chains.

Finding the radical of an algebra of linear transformations

We present a method that reduces the problem of computing the radical of a matrix algebra over an arbitrary field to solving systems of semilinear equations. The complexity of the algorithm, measured in the number of arithmetic operations and the total number of the coefficients passed to an oracle for solving semilinear equations, is polynomial. As an application of the technique we present a simple test for isomorphism of semisimple modules.

Hamming number technique—II. Generation of planar kinematic chains

Synthesis of planar kinematic chains is in general tedious and time consuming. This paper reports a very simple and direct method for the generation of n-link Φ-degree-of-freedom (d.o.f.) chains from ( n − 2)-link and fd.o.f. chains. Hamming number technique reported earlier [A. C. Rao and D. Varada Raju, Mechanism and Machine Theory 26, 55–75 (1991)] for testing isomorphism among kinematic chains is extended to reveal identity and symmetry among the links and joints of a chain. This in turn enables generation of distinct n-link chains directly from each distinct ( n − 2)-link without having to test for degeneration and isomorphism. The formulae suggested will give the number of such chains. Chains with ( n − 1)-links and (f + 1)-d.o.f. are obtained as a by-product. The distinct n-link, fd.o.f. chains obtained from all the basic chains need, however, to be tested for isomorphism among themselves. The advantage of this method can be perceived from the fact that only about 400 10-link chains need be tested to get the already established 230 distinct chains.

Trading Signed Designs and Some New 4-(12, 5, 4) Designs

Variations of the trade-off method exist in the literature of design theory and have been utilized by some authors to produce some t-designs with or without repeated blocks. In this paper we explore a new version of this algorithmic method (i) to produce 20 nonisomorphic and rigid 4-(12,5,4) designs, (ii) to study the spectrum of support sizes of 4-(12,5,4) designs. Along these, we also present a new design invariant for testing isomorphism among designs and a new way of representing t-designs.

Fast generation of cubic graphs

In this paper an efficient algorithm to generate regular graphs with small vertex valency is presented. The running times of a program based on this algorithm and designed to generate cubic graphs are below two natural benchmarks: (a) If N(n) denotes the number of pairwise non-isomorphic cubic graphs with n vertices and T(n) the time needed for generating the list of all these graphs, then T(n)/N(n) decreases gradually for the observed values of n. This suggests that T(n)/N(n) might be uniformly bounded for all n, ignoring the time to write the outputs, but we are unable to prove this and in fact are not confident about it. (b) For programs that generate lists of non-isomorphic objects, but cannot a priori make sure to avoid the generation of isomorphic copies, the time needed to check a randomly ordered list of these objects for being non-isomorphic is a natural benchmark. Since for large lists (n ≥ 22, girth 3) existing graph isomorphism programs take longer to canonically label all of the N(n) graphs than our algorithm takes to generate them, our algorithm is probably faster than any method which does one or more isomorphism test for every graph. © 1996 John Wiley & Sons, Inc.

On the isomorphism problem for a family of cubic metacirculant graphs

In this paper an isomorphism testing algorithm for graphs in the family of all cubic metacirculant graphs with non-empty first symbol S 0 is given. The time complexity of this algorithm is also evaluated.

Graph isomorphism and identification matrices: parallel algorithms

In this paper, we explore some properties of identification matrices and exhibit some uses of identification matrices in studying the graph isomorphism problem, a famous open problem. We show that, given two graphs in the form of a certain identification matrix, isomorphism can be tested efficiently in parallel if at least one matrix satisfies the circular 1s property, and more efficiently in parallel if at least one matrix satisfies the consecutive 1s property. Graphs which have identification matrices satisfying the consecutive 1s property include, among others, proper interval graphs and doubly convex bipartite graphs. The result presented here substantially broadens the class of graphs for which there are known efficient parallel isomorphism testing algorithms.

Isomorphism of chordal (6, 3) graphs

A graph is chordal if it contains no chordless cycles of length at least four and (q, t) if no set of at mostq vertices induces more thant paths of length three. It is known that the isomorphism problem is isomorphism complete for chordal graphs and for (6, 3) graphs. We present polynomial methods to determine the automorphism partition and to test isomorphism of graphs which are both chordal and (6, 3). The approach is based on the study of simplicial partitions of chordal graphs. It is proved that for chordal (6, 3) graphs the automorphism partition coincides with the coarsest regular simplicial partition. This yields anO(n+m logn) isomorphism test.

Stratified graphs for imbedding systems

Two imbeddings of a graph G are considered to be adjacent if the second can be obtained from the first by moving one or both ends of a single edge within its or their respective rotations. Thus, a collection of imbeddings S of G, called a ‘system’, may be represented as a ‘stratified graph’, and denoted SG; the focus here is the case in which S is the collection of all orientable imbeddings. The induced subgraph of SG on the set of imbeddings into the surface of genus k is called the ‘kth stratum’, and the cardinality of that set of imbeddings is called the ‘stratum size’; one may observe that the sequence of stratum sizes is precisely the genus distribution for the graph G. It is known that the genus distribution is not a complete invariant, even when the category of graphs is restricted to be simplicial and 3-connected. However, it is proved herein that the link of each point — that is, the subgraph induced by its neighbors — of SG is a complete isomorphism invariant for the category of graphs whose minimum valence is at least three. This supports the plausibility of a probabilistic approach to graph isomorphism testing by sampling higher-order imbedding distribution data. A detailed structural analysis of stratified graphs is presented.

Star partitions and the graph isomorphism problem

A canonical basis of R n associated with a graph G on n vertices has been defined in [15] in connection with eigenspaces and star partitions of G. The canonical star basis together with eigenvalues of G determines G to an isomorphism. We study algorithms for finding the canonical basis and some of its variations. The emphasis is on the following three special cases; graphs with distinct eigenvalues, graphs with bounded eigenvalue multiplicities and strongly regular graphs. We show that the procedure is reduced in some parts to special cases of some well known combinatorial optimization problems, such as the maximal matching problem. the minimal cut problem, the maximal clique problem etc. This technique provides another proof of a result of L. Babai et al. [2] that isomorphism testing for graphs with bounded eigenvalue multiplicities can be performend in a polynomial time. We show that the canonical basis in strongly regular graphs is related to the graph decomposition into two strongly regular induced su...

Intersection graphs of proper subtrees of unicyclic graphs

AbstractA graph is fraternally oriented iff for every three vertices u, ν, w the existence of the edges u → w and ν → w implies that u and ν are adjacent. A directed unicyclic graph is obtained from a unicyclic graph by orienting the unique cycle clockwise and by orienting the appended subtrees from the cycle outwardly. Two directed subtrees s, t of a directed unicyclic graph are proper if their union contains no (directed or undirected) cycle and either they are disjoint or one of them s has its root r(s) in t and contains all the successors of r(s) in t.In the present paper we prove that G is an intersection graph of a family of proper directed subtrees of a directed unicyclic graph iff it has a fraternal orientation such that for every vertex ν, G(Γinν) is acyclic and G(Γoutν) is the transitive closure of a tree. We describe efficient algorithms for recognizing when such graphs are perfect and for testing isomorphism of proper circular‐arc graphs.

Eccentric sequences and triangle sequences of block designs

For a given block design D, we consider two isomorphism invariants, the eccentric sequences and the triangle sequences of some special graphs of D. Among other results we show that these invariants can often be used effectively, as far as computational complexity is concerned, in isomorphism testing of designs. Several unsolved problems are also proposed.

Isomorphism testing for p-groups

We describe the theoretical and practical details of an algorithm which can be used to decide whether two given presentations for finite p-groups present isomorphic groups. The approach adopted is to construct a canonical presentation for each group. A description of the automorphism group of the p -group is also constructed.

Isomorphism testing for p-groups

We describe the theoretical and practical details of an algorithm which can be used to decide whether two given presentations for finite p-groups present isomorphic groups. The approach adopted is to construct a canonical presentation for each group. A description of the automorphism group of the p -group is also constructed.

Efficient parallel algorithms for bipartite permutation graphs

AbstractIn this paper, we further study the properties of bipartite permutation graphs. We give first efficient parallel algorithms for several problems on bipartite permutation graphs. These problems include transforming a bipartite graph into a strongly ordered one if it is also a permutation graph; testing isomorphism; finding a Hamiltonian path/cycle; solving a variant of the crossing number problem; and others. All these problems can be solved in O(log2n) time with O(n3) processors on a Common CRCW PRAM. We also show that the minimum fill‐in problem for bipartite permutation graphs can be solved efficiently by a randomized parallel algorithm. © 1993 by John Wiley & Sons, Inc.

Loop based pseudo hamming values—I testing isomorphism and rating kinematic chains

Topology of kinematic chains is useful in comparing them from the structural-error point of view and an attempt was made by one of the authors in this direction. The method reported, however, fails to compare the chains which consist of the same number and type of links and joints, ternary-binary, ternary-quaternary, etc. but differ in loop formation only. The present paper supplements the earlier work and makes it complete. Further, comparison of the loop Hamming values of links and chains is expected to be the simplest and positive test for isomorphism. A note on the type of freedom is also included.

Polynomial time algorithms for recognizing and isomorphism testing of cyclic tournaments

In [3] the problem of finding an efficient criterion for isomorphism testing of cyclic graphs was posed. In the context of the theory of computational complexity the problem reduces to that of the existence of a polynomial-time algorithm for recognizing their isomorphism. The main result of the present paper is an algorithm for finding among all tournaments the cyclic ones. For cyclic tournaments generators of the automorphism group and the set of canonical labels are constructed. The running time of the algorithm is bounded by a polynomial function of the number of input tournament vertices. Thus an affirmative answer to the above problem is obtained.

The covering radius of Hadamard codes in odd graphs

The use of odd graphs has been proposed as fault-tolerant interconnection networks. The following problem originated in their design: what is the graphical covering radius of an Hadamard code of length 2 k − 1 and size 2 k − 1 in the odd graph O k ? Of particular interest is the case of k=2 m −1, where we can choose this Hadamard code to be a subcode of the punctured first order Reed-Muller code RM(1, m). We define the w-covering radius of a binary code as the largest Hamming distance from a binary word of Hamming weight w to the code. The above problem amounts to finding the k-covering radius of a (2 k,4 k, k−1) Hadamard code. We find upper and lower bounds on this integer, and determine it for small values of k. Our study suggests a new isomorphism test for Hadamard designs.