scholarly journals The twist for positroids

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Greg Muller ◽  
David E. Speyer
Keyword(s):  
Planar Graphs ◽  
Cluster Structure ◽  
The Other ◽  
Twist Map ◽  
Boundary Measurement ◽  
Bruhat Cells ◽  
International Audience ◽  
Initial Seed ◽  
Plabic Graphs

International audience There are two reasonable ways to put a cluster structure on a positroid variety. In one, the initial seed is a set of Plu ̈cker coordinates. In the other, the initial seed consists of certain monomials in the edge weights of a plabic graph. We will describe an automorphism of the positroid variety, the twist, which takes one to the other. For the big positroid cell, this was already done by Marsh and Scott; we generalize their results to all positroid varieties. This also provides an inversion of the boundary measurement map which is more general than Talaska's, in that it works for all reduced plabic graphs rather than just Le-diagrams. This is the analogue for positroid varieties of the twist map of Berenstein, Fomin and Zelevinsky for double Bruhat cells. Our construction involved the combinatorics of dimer configurations on bipartite planar graphs.

On Cycles and Connectivity in Planar Graphs

1973 ◽  
Vol 16 (2) ◽  
pp. 283-288 ◽  
Author(s):  
M. D. Plummer ◽  
E. L. Wilson
Keyword(s):  
Planar Graphs ◽  
Hamiltonian Cycle ◽  
The Other ◽  
Other Hand

Let G be a graph and ζ(G) be the greatest integer n such that every set of n points in G lies on a cycle [8]. It is clear that ζ(G)≥2 for 2-connected planar graphs. Moreover, it is easy to construct arbitrarily large 2-connected planar graphs for which ζ=2. On the other hand, by a well-known theorem of Tutte [5], [6], if G is planar and 4-connected, it has a Hamiltonian cycle, i.e., ζ(G)=|V(G)| for all 4-connected (and hence for all 5-connected) planar graphs.

scholarly journals Parameterized Problems Related to Seidel's Switching

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Eva Jelinkova ◽  
Ondrej Suchy ◽  
Petr Hlineny ◽  
Jan Kratochvil
Keyword(s):  
Minimum Degree ◽  
The Other ◽  
Complete Graphs ◽  
Maximum Likelihood Decoding ◽  
Induced Subgraph ◽  
Fixed Parameter Tractable ◽  
Graph Theoretic ◽  
Np Completeness ◽  
Fixed Parameter ◽  
International Audience

Graphs and Algorithms International audience Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other by a sequence of switches. In this paper, we continue the study of computational complexity aspects of Seidel's switching, concentrating on Fixed Parameter Complexity. Among other results we show that switching to a graph with at most k edges, to a graph of maximum degree at most k, to a k-regular graph, or to a graph with minimum degree at least k are fixed parameter tractable problems, where k is the parameter. On the other hand, switching to a graph that contains a given fixed graph as an induced subgraph is W [1]-complete. We also show the NP-completeness of switching to a graph with a clique of linear size, and of switching to a graph with small number of edges. A consequence of the latter result is the NP-completeness of Maximum Likelihood Decoding of graph theoretic codes based on complete graphs.

scholarly journals A further correspondence between $(bc,\bar{b})$-parking functions and $(bc,\bar{b})$-forests

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Heesung Shin ◽  
Jiang Zeng
Keyword(s):  
The Other ◽  
Parking Functions ◽  
Fixed Sequence ◽  
Parking Function ◽  
International Audience ◽  
Positive Integers ◽  
Appl Math

International audience For a fixed sequence of $n$ positive integers $(a,\bar{b}) := (a, b, b,\ldots, b)$, an $(a,\bar{b})$-parking function of length $n$ is a sequence $(p_1, p_2, \ldots, p_n)$ of positive integers whose nondecreasing rearrangement $q_1 \leq q_2 \leq \cdots \leq q_n$ satisfies $q_i \leq a+(i-1)b$ for any $i=1,\ldots, n$. A $(a,\bar{b})$-forest on $n$-set is a rooted vertex-colored forests on $n$-set whose roots are colored with the colors $0, 1, \ldots, a-1$ and the other vertices are colored with the colors $0, 1, \ldots, b-1$. In this paper, we construct a bijection between $(bc,\bar{b})$-parking functions of length $n$ and $(bc,\bar{b})$-forests on $n$-set with some interesting properties. As applications, we obtain a generalization of Gessel and Seo's result about $(c,\bar{1})$-parking functions [Ira M. Gessel and Seunghyun Seo, Electron. J. Combin. $\textbf{11}$(2)R27, 2004] and a refinement of Yan's identity [Catherine H. Yan, Adv. Appl. Math. $\textbf{27}$(2―3):641―670, 2001] between an inversion enumerator for $(bc,\bar{b})$-forests and a complement enumerator for $(bc,\bar{b})$-parking functions. Soit $(a,\bar{b}) := (a, b, b,\ldots, b)$ une suite d'entiers positifs. Une $(a,\bar{b})$-fonction de parking est une suite $(p_1, p_2, \ldots, p_n)$ d'entiers positives telle que son réarrangement non décroissant $q_1 \leq q_2 \leq \cdots \leq q_n$ satisfait $q_i \leq a+(i-1)b$ pour tout $i=1,\ldots, n$. Une $(a,\bar{b})$-forêt enracinée sur un $n$-ensemble est une forêt enracinée dont les racines sont colorées avec les couleurs $0, 1, \ldots, a-1$ et les autres sommets sont colorés avec les couleurs $0, 1, \ldots, b-1$. Dans cet article, on construit une bijection entre $(bc,\bar{b})$-fonctions de parking et $(bc,\bar{b})$-forêts avec des des propriétés intéressantes. Comme applications, on obtient une généralisation d'un résultat de Gessel-Seo sur $(c,\bar{1})$-fonctions de parking [Ira M. Gessel and Seunghyun Seo, Electron. J. Combin. $\textbf{11}$(2)R27, 2004] et une extension de l'identité de Yan [Catherine H. Yan, Adv. Appl. Math. $\textbf{27}$(2―3):641―670, 2001] entre l'énumérateur d'inversion de $(bc,\bar{b})$-forêts et l'énumérateur complémentaire de $(bc,\bar{b})$-fonctions de parking.

scholarly journals Vectorised Spreading Activation algorithm for centrality measurement

2011 ◽  
Vol 59 (7) ◽  
pp. 469-476 ◽  
Author(s):  
Alexander Troussov ◽  
František Dařena ◽  
Jan Žižka ◽  
Denis Parra ◽  
Peter Brusilovsky
Keyword(s):  
Network Flows ◽  
State Of The Art ◽  
Network Models ◽  
The Other ◽  
Spreading Activation ◽  
Computationally Efficient ◽  
Current State ◽  
Initial Seed ◽  
Spreading Activation Algorithm ◽  
Computationally Efficient Algorithms

Spreading Activation is a family of graph-based algorithms widely used in areas such as information retrieval, epidemic models, and recommender systems. In this paper we introduce a novel Spreading Activation (SA) method that we call Vectorised Spreading Activation (VSA). VSA algorithms, like “traditional” SA algorithms, iteratively propagate the activation from the initially activated set of nodes to the other nodes in a network through outward links. The level of the node’s activation could be used as a centrality measurement in accordance with dynamic model-based view of centrality that focuses on the outcomes for nodes in a network where something is flowing from node to node across the edges. Representing the activation by vectors allows the use of the information about various dimensionalities of the flow and the dynamic of the flow. In this capacity, VSA algorithms can model multitude of complex multidimensional network flows. We present the results of numerical simulations on small synthetic social networks and multi­dimensional network models of folksonomies which show that the results of VSA propagation are more sensitive to the positions of the initial seed and to the community structure of the network than the results produced by traditional SA algorithms. We tentatively conclude that the VSA methods could be instrumental to develop scalable and computationally efficient algorithms which could achieve synergy between computation of centrality indexes with detection of community structures in networks. Based on our preliminary results and on improvements made over previous studies, we foresee advances and applications in the current state of the art of this family of algorithms and their applications to centrality measurement.

scholarly journals Strong Oriented Chromatic Number of Planar Graphs without Short Cycles

2008 ◽  
Vol Vol. 10 no. 1 ◽  
Author(s):  
Mickael Montassier ◽  
Pascal Ochem ◽  
Alexandre Pinlou
Keyword(s):  
Abelian Group ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Oriented Graph ◽  
Minimal Order ◽  
Outerplanar Graphs ◽  
International Audience

International audience Let M be an additive abelian group. An M-strong-oriented coloring of an oriented graph G is a mapping f from V(G) to M such that f(u) <> j(v) whenever uv is an arc in G and f(v)−f(u) <> −(f(t)−f(z)) whenever uv and zt are two arcs in G. The strong oriented chromatic number of an oriented graph is the minimal order of a group M such that G has an M-strong-oriented coloring. This notion was introduced by Nesetril and Raspaud [Ann. Inst. Fourier, 49(3):1037-1056, 1999]. We prove that the strong oriented chromatic number of oriented planar graphs without cycles of lengths 4 to 12 (resp. 4 or 6) is at most 7 (resp. 19). Moreover, for all i ≥ 4, we construct outerplanar graphs without cycles of lengths 4 to i whose oriented chromatic number is 7.

scholarly journals Prospective for the integration of Blockchain and the IoT for Cluster implementation

2020 ◽  
Vol 16 (3) ◽  
pp. 1-30
Author(s):  
José Ignacio Rodríguez Molano ◽  
Jhonnatan Nicolás Martínez Baracaldo ◽  
Jenny Alexandra Triana Casallas
Keyword(s):  
Cloud Computing ◽  
Internet Of Things ◽  
Present Article ◽  
Cluster Structure ◽  
The Other ◽  
Computer Application ◽  
Structural Level ◽  
Functional Level ◽  
Problem Construction ◽  
Response Network

Introduction: The present article is the result of the investigation and approach to the applications and developments of blockchain and Internet of Things (IoT), developed during the second semester of the year 2019 and first of 2020.Problem: Construction of environments and mediums in a cluster structure that allow companies and institutions to cooperate and compete to achieve efficiency and strengthen grouping. Objective: Integrate blockchain and IoT to develop and present a two-level architecture, from which a support environment is established and a series of functionalities are offered for a cluster implementation.Methodology: Review articles to achieve an approach to blockchain and IoT architecture, configuration and description of structural and functional levels.Results: An architecture with a structural level constituted by a decentralized computer application based on blockchains, a sensory and response network that incorporates IoT technologies and an intermediate component of cloud computing; this, at a functional level that manages to offer users support and help in their activities from modules created with a particular specialty.Conclusion: The structural level furthered the integration of base technologies, blockchain and IoT; on the other hand, the second level of architecture reveals the potential and versatility of these technologies.Originality: Proposal for the implementation of blockchain, IoT and cloud computing in a cluster structure.Limitations: The difficulty of accessing a cluster to perform a test of the architecture in a real environment.

scholarly journals The cluster and dual canonical bases of Z [x_11, ..., x_33] are equal

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Brendon Rhoades
Keyword(s):  
Quantum Group ◽  
Polynomial Ring ◽  
Geometric Model ◽  
Cluster Algebra ◽  
Canonical Basis ◽  
The Other ◽  
Monomial Basis ◽  
Dual Canonical Basis ◽  
Nous Montrons ◽  
International Audience

International audience The polynomial ring $\mathbb{Z}[x_{11}, . . . , x_{33}]$ has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group $U_q(\mathfrak{sl}3(\mathbb{C}))$. On the other hand, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ inherits a basis from the cluster monomial basis of a geometric model of the type $D_4$ cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky. This also provides an explicit factorization of the dual canonical basis elements of $\mathbb{Z}[x_{11}, . . . , x_{33}]$ into irreducible polynomials. L'anneau de polynômes $\mathbb{Z}[x_{11}, . . . , x_{33}]$ a une base appelée base duale canonique, et dont une quantification facilite l'étude des représentations du groupe quantique $U_q(\mathfrak{sl}3(\mathbb{C}))$. D'autre part, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ admet une base issue de la base des monômes d'amas de l'algèbre amassée géométrique de type $D_4$. Nous montrons que ces deux bases sont égales. Ceci prolonge les travaux de Skandera et démontre une conjecture de Fomin et Zelevinsky. Ceci fournit également une factorisation explicite en polynômes irréductibles des éléments de la base duale canonique de $\mathbb{Z}[x_{11}, . . . , x_{33}]$ .

scholarly journals On the k-restricted structure ratio in planar and outerplanar graphs

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Gruia Călinescu ◽  
Cristina G. Fernandes
Keyword(s):  
Approximation Algorithms ◽  
Planar Graphs ◽  
Rates Of Convergence ◽  
Simple Graph ◽  
Maximum Weight ◽  
Outerplanar Graphs ◽  
Weighted Version ◽  
International Audience ◽  
Unweighted Case

Graphs and Algorithms International audience A planar k-restricted structure is a simple graph whose blocks are planar and each has at most k vertices. Planar k-restricted structures are used by approximation algorithms for Maximum Weight Planar Subgraph, which motivates this work. The planar k-restricted ratio is the infimum, over simple planar graphs H, of the ratio of the number of edges in a maximum k-restricted structure subgraph of H to the number edges of H. We prove that, as k tends to infinity, the planar k-restricted ratio tends to 1 = 2. The same result holds for the weighted version. Our results are based on analyzing the analogous ratios for outerplanar and weighted outerplanar graphs. Here both ratios tend to 1 as k goes to infinity, and we provide good estimates of the rates of convergence, showing that they differ in the weighted from the unweighted case.

scholarly journals An improved bound on the largest induced forests for triangle-free planar graphs

2010 ◽  
Vol Vol. 12 no. 1 ◽  
Author(s):  
Lukasz Kowalik ◽  
Borut Luzar ◽  
Riste Skrekovski
Keyword(s):  
Planar Graphs ◽  
Free Graph ◽  
International Audience

International audience We proved that every planar triangle-free graph of order n has a subset of vertices that induces a forest of size at least (71n + 72)/128. This improves the earlier work of Salavatipour (2006). We also pose some questions regarding planar graphs of higher girth.

scholarly journals The cluster and dual canonical bases of Z[x(11), ..., x(33)] are equal

2010 ◽  
Vol Vol. 12 no. 5 (Combinatorics) ◽  
Author(s):  
Brendon Rhoades
Keyword(s):  
Quantum Group ◽  
Polynomial Ring ◽  
Geometric Model ◽  
Cluster Algebra ◽  
Canonical Basis ◽  
The Other ◽  
Monomial Basis ◽  
Type D ◽  
Dual Canonical Basis ◽  
International Audience

Combinatorics International audience The polynomial ring Z[x(11), ..., x(33)] has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group U-q(sl(3) (C)). On the other hand, Z[x(1 1), ... , x(33)] inherits a basis from the cluster monomial basis of a geometric model of the type D-4 cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky.

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