Computing Lengths of Shortest Non-Crossing Paths in Planar Graphs

Given a plane undirected graph $G$ with non-negative edge weights and a set of $k$ terminal pairs on the external face, it is shown in Takahashi et al., (Algorithmica, 16, 1996, pp. 339-357) that the lengths of $k$ non-crossing shortest paths joining the $k$ terminal pairs (if they exist) can be computed in $O(n \log n)$ worst-case time, where $n$ is the number of vertices of $G$. This technique only applies when the union $U$ of the computed shortest paths is a forest. We show that given a plane undirected weighted graph $U$ and a set of $k$ terminal pairs on the external face, it is always possible to compute the lengths of $k$ non-crossing shortest paths joining the $k$ terminal pairs in linear worst-case time, provided that the graph $U$ is the union of $k$ shortest paths, possibly containing cycles. Moreover, each shortest path $\pi$ can be listed in $O(\ell+\ell\log\lceil{\frac{k}{\ell}}\rceil)$, where $\ell$ is the number of edges in $\pi$. As a consequence, the problem of computing multi-terminal distances in a plane undirected weighted graph can always be solved in $O(n \log k)$ worst-case time in the general case.Comment: 17 pages, 11 figure

Network homophily via multi-dimensional extensions of Cantelli's inequality

Homophily is the principle whereby "similarity breeds connections". We give a quantitative formulation of this principle within networks. We say that a network is homophillic with respect to a given labeled partition of its vertices, when the classes of the partition induce subgraphs that are significantly denser than what we expect under a random labeled partition into classes maintaining the same cardinalities (type). This is the recently introduced \emph{random coloring model} for network homophily. In this perspective, the vector whose entries are the sizes of the subgraphs induced by the corresponding classes, is viewed as the observed outcome of the random vector described by picking labeled partitions at random among partitions with the same type.\,Consequently, the input network is homophillic at the significance level $\alpha$ whenever the one-sided tail probability of observing an outcome at least as extreme as the observed one, is smaller than $\alpha$. Clearly, $\alpha$ can also be thought of as a quantifier of homophily in the scale $[0,1]$. Since, as we show, even approximating this tail probability is an NP-hard problem, we resort multidimensional extensions of classical Cantelli's inequality to bound $\alpha$ from above. This upper bound is the homophily index we propose. It requires the knowledge of the covariance matrix of the random vector, which was not previously known within the random coloring model. In this paper we close this gap by computing the covariance matrix of subgraph sizes under the random coloring model. Interestingly, the matrix depends on the input partition only through its type and on the network only through its degrees. Furthermore all the covariances have the same sign and this sign is a graph invariant. Plugging this structure into Cantelli's bound yields a meaningful, easy to compute index for measuring network homophily

Max-flow vitality in undirected unweighted planar graphs

We show a fast algorithm for determining the set of relevant edges in a planar undirected unweighted graph with respect to the maximum flow. This is a special case of the \emph{max flow vitality} problem, that has been efficiently solved for general undirected graphs and $st$-planar graphs. The \emph{vitality} of an edge of a graph with respect to the maximum flow between two fixed vertices $s$ and $t$ is defined as the reduction of the maximum flow caused by the removal of that edge. In this paper we show that the set of edges having vitality greater than zero in a planar undirected unweighted graph with $n$ vertices, can be found in $O(n \log n)$ worst-case time and $O(n)$ space.Comment: 9 pages, 4 figure

Clostridium botulinum spores and toxin in mascarpone cheese and other milk products

A total of 1,017 mascarpone cheese samples, collected at retail, were analyzed for Clostridium botulinum spores and toxin, aerobic mesophilic spore counts, as well as pH, a(w) (water activity), and Eh (oxidation-reduction potential). In addition 260 samples from other dairy products were also analyzed for spores and botulinum toxin. Experiments were carried out on naturally and artificially contaminated mascarpone to investigate the influence of different temperature conditions on toxin production by C. botulinum. Three hundred and thirty-one samples (32.5%) of mascarpone were positive for botulinal spores, and 7 (0.8%) of the 878 samples produced at the plant involved in an outbreak of foodborne botulism also contained toxin type A. The chemical-physical parameters (pH, a(w), Eh) of all samples were compatible with C. botulinum growth and toxinogenesis. Of the other milk products, 2.7% were positive for C. botulinum spores. Growth and toxin formation occurred in naturally and experimentally contaminated mascarpone samples after 3 and 4 days of incubation at 28 degrees C, respectively

Identification of Novel Linear Megaplasmids Carrying a ß-Lactamase Gene in Neurotoxigenic Clostridium butyricum Type E Strains

Since the first isolation of type E botulinum toxin-producing Clostridium butyricum from two infant botulism cases in Italy in 1984, this peculiar microorganism has been implicated in different forms of botulism worldwide. By applying particular pulsed-field gel electrophoresis run conditions, we were able to show for the first time that ten neurotoxigenic C. butyricum type E strains originated from Italy and China have linear megaplasmids in their genomes. At least four different megaplasmid sizes were identified among the ten neurotoxigenic C. butyricum type E strains. Each isolate displayed a single sized megaplasmid that was shown to possess a linear structure by ATP-dependent exonuclease digestion. Some of the neurotoxigenic C. butyricum type E strains possessed additional smaller circular plasmids. In order to investigate the genetic content of the newly identified megaplasmids, selected gene probes were designed and used in Southern hybridization experiments. Our results revealed that the type E botulinum neurotoxin gene was chromosome-located in all neurotoxigenic C. butyricum type E strains. Similar results were obtained with the 16S rRNA, the tetracycline tet(P) and the lincomycin resistance protein lmrB gene probes. A specific mobA gene probe only hybridized to the smaller plasmids of the Italian C. butyricum type E strains. Of note, a ß-lactamase gene probe hybridized to the megaplasmids of eight neurotoxigenic C. butyricum type E strains, of which seven from clinical sources and the remaining one from a food implicated in foodborne botulism, whereas this ß-lactam antibiotic resistance gene was absent form the megaplasmids of the two soil strains examined. The widespread occurrence among C. butyricum type E strains associated to human disease of linear megaplasmids harboring an antibiotic resistance gene strongly suggests that the megaplasmids could have played an important role in the emergence of C. butyricum type E as a human pathogen

Non-crossing shortest paths in undirected unweighted planar graphs in linear time

Given a set of terminal pairs on the external face of an undirected unweighted planar graph, we give a linear-time algorithm for computing the union of non-crossing shortest paths joining each terminal pair, if such paths exist. This allows us to compute distances between each terminal pair, within the same time bound. We also give a novel concept of incremental shortest path subgraph of a planar graph, i.e., a partition of the planar embedding in subregions that preserve distances, that can be of interest itself

Multi-Terminal Shortest Paths in Unit-Weight Planar Graphs in Linear Time

Given a set of terminal pairs on the external face of a planar graph with unit edge weights, we give a linear-time algorithm to compute a set of non-crossing shortest paths joining each terminal pair, if it exists

Max Flow Vitality of Edges and Vertices in Undirected Planar Graphs

We study the problem of computing the vitality of edges and vertices with respect to $st$-max flow in undirected planar graphs, where the vitality of an edge/vertex in a graph with respect to max flow between two fixed vertices $s,t$ is defined as the max flow decrease when the edge/vertex is removed from the graph. We show that a $delta$ additive approximation of the vitality of all edges with capacity at most $c$ can be computed in $O(rac{c}{delta}n +n log log n)$ time, where $n$ is the size of the graph. A similar result is given for the vitality of vertices. All our algorithms work in $O(n)$ space

On the Boolean dimension of spherical orders

It is well known that if a planar order P is bounded, i.e. has only one minimum and one maximum, then the dimension of P (LD(P)) is at most 2, and if we remove the restriction that P has only one maximum then LD(P) less than or equal to 3. However, the dimension of a bounded order drawn on the sphere can be arbitrarily large. The Boolean dimension BD(P) of a poset P is the minimum number of linear orders such that the order relation of P can be written as some Boolean combination of the linear orders. We show that the Boolean dimension of bounded spherical orders is never greater than 4, and is not greater than 5 in the case the poset has more than one maximal element, but only one minimum. These results are obtained by a characterization of spherical orders in terms of containment between circular arcs

Small

stretch spanners on dynamic graphs