Factoring cardinal product graphs in polynomial time

AbstractIn this paper a polynomial algorithm for the prime factorization of finite, connected nonbipartite graphs with respect to the cardinal product is presented. This algorithm also decomposes finite, connected graphs into their prime factors with respect to the strong product and provides the basis for a new proof of the uniqueness of the prime factorization of finite, connected nonbipartite graphs with respect to the cardinal product. Furthermore, some of the consequences of these results and several open problems are discussed

Direct Product Primality Testing of Graphs is GI-hard

We investigate the computational complexity of the graph primality testing problem with respect to the direct product (also known as Kronecker, cardinal or tensor product). In [1] Imrich proves that both primality testing and a unique prime factorization can be determined in polynomial time for (finite) connected and nonbipartite graphs. The author states as an open problem how results on the direct product of nonbipartite, connected graphs extend to bipartite connected graphs and to disconnected ones. In this paper we partially answer this question by proving that the graph isomorphism problem is polynomial-time many-one reducible to the graph compositeness testing problem (the complement of the graph primality testing problem). As a consequence of this result, we prove that the graph isomorphism problem is polynomial-time Turing reducible to the primality testing problem. Our results show that connectedness plays a crucial role in determining the computational complexity of the graph primality testing problem

Some properties on the lexicographic product of graphs obtained by monogenic semigroups

In (Das et al. in J. Inequal. Appl. 2013:44, 2013), a new graph Gamma (S-M) on monogenic semigroups S-M (with zero) having elements {0, x, x(2), x(3),..., x(n)} was recently defined. The vertices are the non-zero elements x, x(2), x(3),..., x(n) and, for 1 <= i, j <= n, any two distinct vertices x(i) and x(j) are adjacent if x(i)x(j) = 0 in S-M. As a continuing study, in an unpublished work, some well-known indices (first Zagreb index, second Zagreb index, Randic index, geometric-arithmetic index, atom-bond connectivity index, Wiener index, Harary index, first and second Zagreb eccentricity indices, eccentric connectivity index, the degree distance) over Gamma (S-M) were investigated by the same authors of this paper. In the light of the above references, our main aim in this paper is to extend these studies to the lexicographic product over Gamma (S-M). In detail, we investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the lexicographic product of any two (not necessarily different) graphs Gamma (S-M(1)) and Gamma (S-M(2)).Selçuk ÜniversitesiSungkyunkwan University (BK21

A Heuristic for Direct Product Graph Decomposition

In this paper we describe a heuristic for decomposing a directed graph into factors according to the direct product (also known as Kronecker, cardinal or tensor product). Given a directed, unweighted graph G with adjacency matrix Adj(G), our heuristic aims at identifying two graphs G 1 and G 2 such that G = G 1 × G 2 , where G 1 × G 2 is the direct product of G 1 and G 2 . For undirected, connected graphs it has been shown that graph decomposition is “at least as difficult” as graph isomorphism; therefore, polynomial-time algorithms for decomposing a general directed graph into factors are unlikely to exist. Although graph factorization is a problem that has been extensively investigated, the heuristic proposed in this paper represents – to the best of our knowledge – the first computational approach for general directed, unweighted graphs. We have implemented our algorithm using the MATLAB environment; we report on a set of experiments that show that the proposed heuristic solves reasonably- sized instances in a few seconds on general-purpose hardware. Although the proposed heuristic is not guaranteed to find a factorization, even if one exists; however, it always succeeds on all the randomly-generated instances used in the experimental evaluation

On Cartesian skeletons of graphs,

Abstract Under suitable conditions of connectivity or non-bipartiteness, each of the three standard graph products (the Cartesian product, the direct product and the strong product) satisfies the unique prime factorization property, and there are polynomial algorithms to determine the prime factors. This is most easily proved for the Cartesian product. For the other products, current proofs involve a notion of a Cartesian skeleton which transfers their multiplication properties to the Cartesian product. The present article introduces simplified definitions of Cartesian skeletons for the direct and strong products, and provides new, fast and transparent algorithms for their construction. Since the complexity of the prime factorization of the direct and the strong product is determined by the complexity of the construction of the Cartesian skeleton, the new algorithms also improve the complexity of the prime factorizations of graphs with respect to the direct and the strong product. We indicate how these simplifications fit into the existing literature

The Topology of Evolutionary Biology

Central notions in evolutionary biology are intrinsically topological. This claim is maybe most obvious for the discontinuities associated with punctuated equilibria. Recently, a mathematical framework has been developed that derives the concepts of phenotypic characters and homology from the topological structure of the phenotype space. This structure in turn is determined by the genetic operators and their interplay with the properties of the genotype-phenotype map

Hyperbolicity of direct products of graphs

It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G(1) x G(2) is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs).This work was supported in part by four grants from Ministerio de Economía y Competititvidad (MTM2012-30719, MTM2013-46374-P, MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain