Bipartite powers of k-chordal graphs

Let k be an integer and k \geq 3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if G^m is chordal then so is G^{m+2}. Brandst\"adt et al. in [Andreas Brandst\"adt, Van Bang Le, and Thomas Szymczak. Duchet-type theorems for powers of HHD-free graphs. Discrete Mathematics, 177(1-3):9-16, 1997.] showed that if G^m is k-chordal, then so is G^{m+2}. Powering a bipartite graph does not preserve its bipartitedness. In order to preserve the bipartitedness of a bipartite graph while powering Chandran et al. introduced the notion of bipartite powering. This notion was introduced to aid their study of boxicity of chordal bipartite graphs. Given a bipartite graph G and an odd positive integer m, we define the graph G^{[m]} to be a bipartite graph with V(G^{[m]})=V(G) and E(G^{[m]})={(u,v) | u,v \in V(G), d_G(u,v) is odd, and d_G(u,v) \leq m}. The graph G^{[m]} is called the m-th bipartite power of G. In this paper we show that, given a bipartite graph G, if G is k-chordal then so is G^{[m]}, where k, m are positive integers such that k \geq 4 and m is odd.Comment: 10 page

07211 Abstracts Collection -- Exact, Approximative, Robust and Certifying Algorithms on Particular Graph Classes

From May 20 to May 25, 2007, the Dagstuhl Seminar 07211 ``Exact, Approximative, Robust and Certifying Algorithms on Particular Graph Classes\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Hadamard-Hitchcock decompositions: identifiability and computation

A Hadamard-Hitchcock decomposition of a multidimensional array is a decomposition that expresses the latter as a Hadamard product of several tensor rank decompositions. Such decompositions can encode probability distributions that arise from statistical graphical models associated to complete bipartite graphs with one layer of observed random variables and one layer of hidden ones, usually called restricted Boltzmann machines. We establish generic identifiability of Hadamard-Hitchcock decompositions by exploiting the reshaped Kruskal criterion for tensor rank decompositions. A flexible algorithm leveraging existing decomposition algorithms for tensor rank decomposition is introduced for computing a Hadamard-Hitchcock decomposition. Numerical experiments illustrate its computational performance and numerical accuracy.Comment: 25 pages, 3 figure

Fast approximation of centrality and distances in hyperbolic graphs

We show that the eccentricities (and thus the centrality indices) of all vertices of a $\delta$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $c\delta$, i.e., after a linear time preprocessing, for every vertex $v$ of $G$ one can compute in $O(1)$ time an estimate $\hat{e}(v)$ of its eccentricity $ecc_G(v)$ such that $ecc_G(v)\leq \hat{e}(v)\leq ecc_G(v)+ c\delta$ for a small constant $c$. We prove that every $\delta$-hyperbolic graph $G$ has a shortest path tree, constructible in linear time, such that for every vertex $v$ of $G$, $ecc_G(v)\leq ecc_T(v)\leq ecc_G(v)+ c\delta$. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of $G$, the smaller its eccentricity is. We also show that the distance matrix of $G$ with an additive one-sided error of at most $c'\delta$ can be computed in $O(|V|^2\log^2|V|)$ time, where $c'< c$ is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author

Study of the Driving Cycle for Heavy Duty Trucks in Hilly Terrain and Its Effect on Calculated Emissions, and Comparison of Two Mobile Emission Models

Vehicle emissions were estimated using MOVES2010a and MOBILE6.2 for a Pittsburgh case study involving a modal shift in freight transportion. MOVES2010a (hereafter referred to as MOVES) is currently the USEPA official mobile source emissions computer model; it replaced the older model, MOBILE6.2. Changing the method of hauling freight from highway to waterway is the transport modal shift. Results from this part of the study showed that emission estimates for all vehicle types using MOVES were higher than emissions estimated using MOBILE6.2/NMIM for CO, NOX, PM10, PM2.5, and VOC, but emissions were lower for CO2 and NH3 using MOVES relative to MOBILE6.2. For heavy-heavy duty diesel (HHDD) vehicles, higher emissions were estimated using MOVES for all pollutants except for NH3 when compared to MOBILE6.2. The largest difference between the two models was seen in PM10 and PM2.5. The second part of this dissertation focused on driving cycles for HHDD vehicles in hilly terrain and its effect on emissions. The MOVES model incorporates 12 default driving schedules for HHDD vehicles. Each driving schedule represents different average vehicle speeds, which tend to over generalize the driving patterns for these vehicles in hilly terrain. The characteristics of HHDD vehicle driving cycles were analyzed by using actual GPS speed and terrain data from driving activity that occurred on a section of the Federal Interstate to demonstrate possible drawbacks of default driving schedules in the current version of MOVES. Profiles of speed versus time as well as road grades were constructed to validate this. Emissions were calculated using a MOVES’ operating mode approach. Results showed that a wider range of speeds and higher scaled tractive power occurred in the driving cycles constructed from the real activity data in hilly terrain than the MOVES default driving schedules. NOX, PM2.5, and THC emissions and total energy consumption calculated using the synthetic driving cycles of the trucks with grades, associated with the hilly terrain, were 7.6%, 14%, 3%, and 11%, respectively, higher than when using the MOVES default driving schedules at the same average speed (63.9 mph) for 0.3% average road grade. On the other hand, CO emissions were 3.4% lower for the synthetic driving cycles. More analyses associated with the driving cycles were presented in this dissertation, and recommendations were made regarding an improvement of default driving schedules in MOVES as well