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The Directional Discrete Cosine Transform
Published in Humberto Ochoa-Domínguez, K. R. Rao, Discrete Cosine Transform, 2019
Humberto Ochoa-Domínguez, K. R. Rao
If an edgeEdge is drawn from vertex to itself, it is called a loop. If there is only one edge connected to the vertex, the vertex is called pendent vertex and if no edges are formed at a vertexVertex, the vertex is an isolated vertex. Figure 7.5 shows the loop at vertex v2 with edge (v2,v2), the isolated vertex v6 and the pendent vertex v5.
Domain-Modeling Techniques
Published in Alexandru Telea, Data Visualization, 2014
A progressive mesh consists of a base mesh, created by a sequence of edge collapse operations on a polygonal mesh, and a sequence of vertex split operations [Hoppe 97]. A split is the dual of a collapse, and it replaces a vertex with two edge-connected vertices, creating an extra vertex and two extra triangles. The base mesh can be exactly transformed into the original model via splits, and the model is transformed into the base mesh via collapses. Intermediate versions correspond to progressive simplifications.
Flows and colorings
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
Delia Garijo, Andrew Goodall, Jaroslav Nešetřil
Grötzsch's theorem[588] states that every triangle-free loopless planar graph is 3-colorable. By coloring-flow duality, this is equivalent to the statement that every 4-edge-connected planar graph has a nowhere-zero 3-flow. Tutte's 3-flow conjecture (first popularized in the 1970s [162] as formulated by Tutte in 1972) asserts that this is still true without the assumption of planarity.
Sensor fault detection and isolation via networked estimation: rank-deficient dynamical systems
Published in International Journal of Control, 2022
M. Doostmohammadian, H. Zarrabi, T. Charalambous
Following the definitions in Section 2, the concept of observational-equivalence is in close relation with q-redundant observability. We are interested to design resilient networked estimators to tolerate isolation/removal of faulty sensors (or failed sensors). In this direction, Algorithm 3 designs q-redundant observable estimators. q number of detected faulty sensors (via the FDI logic in Algorithm 1) can be isolated/removed, while the remaining sensor network preserves distributed observability. Note that q is limited by the minimum size of observationally-equivalent sets (i.e. contractions and parent SCCs), and this min size is defined by in Algorithm 3. Similarly, one can extend the results to design q-edge-connected networks which remain SC after removal of (less than or equal to) q number of links. This is referred to as survivable network design (Lau et al., 2009; Sadeghi & Fan, 2019; Umsonst, 2019) and particularly is related to the connectivity requirement of the Type-β sensors. In other words, designing a q-edge-connected sensor network ensures strong-connectivity after removal of (up to) q links (or q lost-connectivity/missing-packets), which guarantees the connectivity requirement sensors over . Similarly, for , one can add more links from observationally-equivalent α sensors.
Super extra edge-connectivity in regular networks with edge faults
Published in International Journal of Parallel, Emergent and Distributed Systems, 2021
The classical edge-connectivity tacitly assume that all edges that are incident to the same vertex can potentially fail simultaneously. This is practically impossible in large network applications [4]. To address this deficiency, Fbrega and Fiol [5,6] introduced extra edge-connectivity to meet the increasing need of a more accurate measure of the reliability of large-scale networks. For a non-negative integer h, the h-extra edge-connectivity of a connected graph G, if any, denoted by , is the minimum cardinality of an edge set S (named as h-extra edge-cut), whose deletion yields a disconnected graph with each remaining component of G−S having more than h vertices. Note that is the classical edge-connectivity. A more refined measurement is the super h-extra edge-connectivity. A graph G is called super h-extra edge-connected (or simply super-), if every minimum h-extra edge-cut isolates at least one component of cardinality h + 1. For convenience, we use , super edge-connected (or simply super-λ) and super extra edge-connected (or simply super-) instead of , super 0-extra edge-connected and super 1-extra edge-connected in the rest of the paper, respectively.
A methodology for leak detection in water distribution networks using graph theory and artificial neural network
Published in Urban Water Journal, 2020
Mohammadreza Shekofteh, Mohammadreza Jalili Ghazizadeh, Jafar Yazdi
After calculating the weights and distances for all nodes, the following steps are taken to calculate the score for each edge: Specify the leaf nodes (l) in the network (The latest nodes of a graph for which there is only one path to the node (r), are called leaf nodes).The score of the edge that connects each leaf node with its neighbor (i) is assigned.Other edges score, starting from the farthest edge from the reference node (analysis is bottom-top). To calculate the score of the edge connected to the nodes i and j (j is farther from r than i), assign a score that is the sum of the scores on the neighboring edges below it (i.e. those with which it shares a common vertex), plus 1, and then the result is multiplied by .Repeat from step 3 until node r is reached.