A Necessary and Sufficient Condition for Graph Matching to be equivalent to Clique Search

This paper formulates a necessary and sufficient condition for a generic graph matching problem to be equivalent to the maximum vertex and edge weight clique problem in a derived association graph. The consequences of this results are threefold: first, the condition is general enough to cover a broad range of practical graph matching problems; second, a proof to establish equivalence between graph matching and clique search reduces to showing that a given graph matching problem satisfies the proposed condition;\ud and third, the result sets the scene for generic continuous solutions for a broad range of graph matching problems. To illustrate the mathematical framework, we apply it to a number of graph matching problems, including the problem of determining the graph edit distance

Structural Properties of the Caenorhabditis elegans Neuronal Network

Despite recent interest in reconstructing neuronal networks, complete wiring diagrams on the level of individual synapses remain scarce and the insights into function they can provide remain unclear. Even for Caenorhabditis elegans, whose neuronal network is relatively small and stereotypical from animal to animal, published wiring diagrams are neither accurate nor complete and self-consistent. Using materials from White et al. and new electron micrographs we assemble whole, self-consistent gap junction and chemical synapse networks of hermaphrodite C. elegans. We propose a method to visualize the wiring diagram, which reflects network signal flow. We calculate statistical and topological properties of the network, such as degree distributions, synaptic multiplicities, and small-world properties, that help in understanding network signal propagation. We identify neurons that may play central roles in information processing and network motifs that could serve as functional modules of the network. We explore propagation of neuronal activity in response to sensory or artificial stimulation using linear systems theory and find several activity patterns that could serve as substrates of previously described behaviors. Finally, we analyze the interaction between the gap junction and the chemical synapse networks. Since several statistical properties of the C. elegans network, such as multiplicity and motif distributions are similar to those found in mammalian neocortex, they likely point to general principles of neuronal networks. The wiring diagram reported here can help in understanding the mechanistic basis of behavior by generating predictions about future experiments involving genetic perturbations, laser ablations, or monitoring propagation of neuronal activity in response to stimulation

Sublinear Distance Labeling

A distance labeling scheme labels the $n$ nodes of a graph with binary strings such that, given the labels of any two nodes, one can determine the distance in the graph between the two nodes by looking only at the labels. A $D$-preserving distance labeling scheme only returns precise distances between pairs of nodes that are at distance at least $D$ from each other. In this paper we consider distance labeling schemes for the classical case of unweighted graphs with both directed and undirected edges. We present a $O(\frac{n}{D}\log^2 D)$ bit $D$-preserving distance labeling scheme, improving the previous bound by Bollob\'as et. al. [SIAM J. Discrete Math. 2005]. We also give an almost matching lower bound of $\Omega(\frac{n}{D})$. With our $D$-preserving distance labeling scheme as a building block, we additionally achieve the following results: 1. We present the first distance labeling scheme of size $o(n)$ for sparse graphs (and hence bounded degree graphs). This addresses an open problem by Gavoille et. al. [J. Algo. 2004], hereby separating the complexity from distance labeling in general graphs which require $\Omega(n)$ bits, Moon [Proc. of Glasgow Math. Association 1965]. 2. For approximate $r$-additive labeling schemes, that return distances within an additive error of $r$ we show a scheme of size $O\left ( \frac{n}{r} \cdot\frac{\operatorname{polylog} (r\log n)}{\log n} \right )$ for $r \ge 2$. This improves on the current best bound of $O\left(\frac{n}{r}\right)$ by Alstrup et. al. [SODA 2016] for sub-polynomial $r$, and is a generalization of a result by Gawrychowski et al. [arXiv preprint 2015] who showed this for $r=2$.Comment: A preliminary version of this paper appeared at ESA'1