Graph Isomorphism and Identification Matrices: Sequential Algorithms

AbstractA number of properties on identification matrices are presented here. For example, we prove that adjacency matrices are identification matrices for all bipartite graphs. We also study the application of the theory of identification matrices to solving the graph isomorphism problem, a famous open problem. We show that, given two graphs represented by two identification matrices with respect to a certain relation, isomorphism can be decided efficiently if at least one matrix satisfies the consecutive 1's property or a relaxed property thereof. Graphs which have identification matrices satisfying the consecutive 1's property include, among others, proper interval graphs and doubly convex bipartite graphs. This work leads to the first efficient isomorphism testing algorithms for certain classes of graphs and more efficient algorithms for some other classes of graphs. The algorithms for some classes of graphs including convex bipartite graphs run in linear time and are optimal

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

Dimers, Tilings and Trees

Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees on the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to ``discrete analytic functions'' on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we classify the resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure

Expander Graph and Communication-Efficient Decentralized Optimization

In this paper, we discuss how to design the graph topology to reduce the communication complexity of certain algorithms for decentralized optimization. Our goal is to minimize the total communication needed to achieve a prescribed accuracy. We discover that the so-called expander graphs are near-optimal choices. We propose three approaches to construct expander graphs for different numbers of nodes and node degrees. Our numerical results show that the performance of decentralized optimization is significantly better on expander graphs than other regular graphs.Comment: 2016 IEEE Asilomar Conference on Signals, Systems, and Computer