Improved random graph isomorphism

Canonical labeling of a graph consists of assigning a unique label to each vertex such that the labels are invariant under isomorphism. Such a labeling can be used to solve the graph isomorphism problem. We give a simple, linear time, high probability algorithm for the canonical labeling of a G(n,p) random graph for p[ω(ln⁴n/nlnlnn),1−ω(ln⁴n/nlnlnn)]. Our result covers a gap in the range of p in which no algorithm was known to work with high probability. Together with a previous result by Bollobás, the random graph isomorphism problem can be solved efficiently for p[Θ(lnn/n),1−Θ(lnn/n)].     

Two graph isomorphism polytopes

The convex hull ψ_n,n of certain (n!)² tensors was considered recently in connection with graph isomorphism. We consider the convex hull ψ_n of the n! diagonals among these tensors. We show: 1. The polytope ψ_n is a face of ψ_n,n. 2. Deciding if a graph G has a subgraph isomorphic to H reduces to optimization over ψ_n. 3. Optimization over ψ_n reduces to optimization over ψ_n,n. In particular, this implies that the subgraph isomorphism problem reduces to optimization over ψ_n,n.     

On graphs whose Laplacian matrix’s multipartite separability is invariant under graph isomorphism

Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density matrix and the graph theoretical properties of the underlying graph. One important aspect of density matrices is their entanglement properties, which are responsible for many nonintuitive physical phenomena. The entanglement property of normalized Laplacian matrices is in general not invariant under graph isomorphism. In recent papers, graphs were identified whose entanglement and separability properties are invariant under isomorphism. The purpose of this note is to completely characterize the set of graphs whose separability is invariant under graph isomorphism. In particular, we show that this set consists of K_2,2 and its complement, all complete graphs and no other graphs.     

Complexity results in graph reconstruction

We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertex-deleted or edge-deleted subgraphs (in particular, deck checking (DC) and legitimate deck (LD) problems). We show that these problems are closely related for all amounts c?1 of deletion:

(1)

, , , and .

(2)

For all k?2, and .

(3)

For all k?2, .

(4)

.

(5)

For all k?2, .

For many of these results, even the c=1 case was not previously known.Similar to the definition of reconstruction numbers vrn_∃(G) [F. Harary, M. Plantholt, The graph reconstruction number, J. Graph Theory 9 (1985) 451-454] and ern_∃(G) (see [J. Lauri, R. Scapellato Topics in Graph Automorphism and Reconstruction, London Mathematical Society, Cambridge University Press, Cambridge, 2003, p. 120]), we introduce two new graph parameters, vrn_∀(G) and ern_∀(G), and give an example of a family {G_n}_n?4 of graphs on n vertices for which vrn_∃(G_n)<vrn_∀(G_n). For every k?2 and n?1, we show that there exists a collection of k graphs on (2^k-1+1)n+k vertices with 2ⁿ 1-vertex-preimages, i.e., one has families of graph collections whose number of 1-vertex-preimages is huge relative to the size of the graphs involved.     

Parameterized graph cleaning problems

We investigate the Induced Subgraph Isomorphism problem with non-standard parameterization, where the parameter is the difference |V(G)|−|V(H)| with H and G being the smaller and the larger input graph, respectively. Intuitively, we can interpret this problem as “cleaning” the graph G, regarded as a pattern containing extra vertices indicating errors, in order to obtain the graph H representing the original pattern. We show fixed-parameter tractability of the cases where both H and G are planar and H is 3-connected, or H is a tree and G is arbitrary. We also prove that the problem when H and G are both 3-connected planar graphs is NP-complete without parameterization.     

Groups with the same non-commuting graph

The non-commuting graph Γ_G of a non-abelian group G is defined as follows. The vertex set of Γ_G is G−Z(G) where Z(G) denotes the center of G and two vertices x and y are adjacent if and only if xy≠yx. It has been conjectured that if G and H are two non-abelian finite groups such that Γ_G≅Γ_H, then |G|=|H| and moreover in the case that H is a simple group this implies G≅H. In this paper, our aim is to prove the first part of the conjecture for all the finite non-abelian simple groups H. Then for certain simple groups H, we show that the graph isomorphism Γ_G≅Γ_H implies G≅H.     

Equivalent formulae for the supremum and stability of weighted pseudoinverses

During recent decades, there have been a great number of research articles studying interior-point methods for solving problems in mathematical programming and constrained optimization. Stewart and O'Leary obtained an upper bound for scaled pseudoinverses of a matrix where is a set of diagonal positive definite matrices. We improved their results to obtain the supremum of scaled pseudoinverses and derived the stability property of scaled pseudoinverses. Forsgren further generalized these results to derive the supremum of weighted pseudoinverses where is a set of diagonally dominant positive semidefinite matrices, by using a signature decomposition of weighting matrices and by applying the Binet-Cauchy formula and Cramer's rule for determinants. The results are also extended to equality constrained linear least squares problems. In this paper we extend Forsgren's results to a general complex matrix to establish several equivalent formulae for , where is a set of diagonally dominant positive semidefinite matrices, or a set of weighting matrices arising from solving equality constrained least squares problems. We also discuss the stability property of these weighted pseudoinverses.

     

A Branch-and-Cut algorithm for graph coloring

In this paper a Branch-and-Cut algorithm, based on a formulation previously introduced by us, is proposed for the Graph Coloring Problem. Since colors are indistinguishable in graph coloring, there may typically exist many different symmetrical colorings associated with a same number of colors. If solutions to an integer programming model of the problem exhibit that property, the Branch-and-Cut method tends to behave poorly even for small size graph coloring instances. Our model avoids, to certain extent, that bottleneck. Computational experience indicates that the results we obtain improve, in most cases, on those given by the well-known exact solution graph coloring algorithm Dsatur.     

A memetic algorithm for graph coloring

Given an undirected graph G=(V,E)$G=(V\text{,}E)$ with a set V of vertices and a set E of edges, the graph coloring problem consists of partitioning all vertices into k independent sets and the number of used colors k is minimized. This paper presents a memetic algorithm (denoted by MACOL) for solving the problem of graph coloring. The proposed MACOL algorithm integrates several distinguished features such as an adaptive multi-parent crossover (AMPaX) operator and a distance-and-quality based replacement criterion for pool updating. The proposed algorithm is evaluated on the DIMACS challenge benchmarks and computational results show that the proposed MACOL algorithm achieves highly competitive results, compared with 11 state-of-the-art algorithms. The influence of some ingredients of MACOL on its performance is also analyzed.     

Computational complexity of reconstruction and isomorphism testing for designs and line graphs

Graphs with high symmetry or regularity are the main source for experimentally hard instances of the notoriously difficult graph isomorphism problem. In this paper, we study the computational complexity of isomorphism testing for line graphs of t-(v,k,λ) designs. For this class of highly regular graphs, we obtain a worst-case running time of O(vlogv+O(1)) for bounded parameters t, k, λ.In a first step, our approach makes use of the Babai-Luks algorithm to compute canonical forms of t-designs. In a second step, we show that t-designs can be reconstructed from their line graphs in polynomial-time. The first is algebraic in nature, the second purely combinatorial. For both, profound structural knowledge in design theory is required. Our results extend earlier complexity results about isomorphism testing of graphs generated from Steiner triple systems and block designs.     

A graph search algorithm for indoor pursuit/evasion

Using concepts from both robotics and graph theory, we formulate the problem of indoor pursuit/evasion in terms of searching the nodes of a graph for a mobile evader. We present the IGNS (Iterative Greedy Node Search) algorithm, which performs offline guaranteed search (i.e. no matter how the evader moves, it will eventually be captured). Furthermore, the algorithm produces an internal search (the searchers move only along the edges of the graph; “teleporting” is not used) and exploits non-monotonicity, extended visibility and finite evader speed to reduce the number of searchers required to clear an environment. We present search experiments for several indoor environments, in all of which the algorithm succeeds in clearing the graph (i.e. capturing the evader).     

A cutting plane algorithm for graph coloring

We present an approach based on integer programming formulations of the graph coloring problem. Our goal is to develop models that remove some symmetrical solutions obtained by color permutations. We study the problem from a polyhedral point of view and determine some families of facets of the 0/1-polytope associated with one of these integer programming formulations. The theoretical results described here are used to design an efficient Cutting Plane algorithm.     

Testing set proportionality and the Ádám isomorphism of circulant graphs

Given two k element subsets , we give a quasi-linear algorithm to either find such that S=λT or prove that no such λ exists.This question is closely related to isomorphism testing of circulant graphs and has recently been studied in the literature.     

A new algorithm for digraph isomorphism

A new and efficient procedure for testing a pair of digraphs for isomorphism is developed. It is based on conducting a depth-first search on one of the digraphs followed by a systematic matching of edges using backtracking with very effective pruning. It is proved that for digraphs (ofn vertices) the expected time complexity of this procedure isO(n ^logn ). This theoretical result is verified empirically on more than 300 large random digraphs. This procedure is shown to be more efficient than any of the existing general isomorphism procedures.     

A realization of graph associahedra

Given any finite graph, we offer a simple realization of its corresponding graph associahedron polytope using integer coordinates.     

On finite groups with the cayley isomorphism property

Let G be a finite group, and let Cay(G, S) be a Cayley digraph of G. If, for all T ⊂ G, Cay(G, S) ≅ Cay(G, T) implies S^α = T for some α ∈ Aut(G), then Cay(G, S) is called a CI-graph of G. For a group G, if all Cayley digraphs of valency m are CI-graphs, then G is said to have the m-DCI property; if all Cayley graphs of valency m are CI-graphs, then G is said to have the m-CI property. It is shown that every finite group of order greater than 2 has a nontrivial CI-graph, and all finite groups with the m-CI property and with the m-DCI property are characterized for small values of m. A general investigation is made of the structure of Sylow subgroups of finite groups with the m-DCI property and with the m-CI property for large values of m. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 21–31, 1998     

On potential isomorphism and non-structure

We show in the paper that for any non-classifiable countable theory T there are non-isomorphic models and that can be forced to be isomorphic without adding subsets of small cardinality. By making suitable cardinal arithmetic assumptions we can often preserve stationary sets as well. We also study non-structure theorems relative to the Ehrenfeucht-Fraïssé game. The research of the first and second author was partially supported by Academy of Finland grant 40734 Mathematics Subject Classification (2000): 03C55, 03C45