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A note on fragile graphs

Guantao Chen, Xingxing Yu
2002 Discrete Mathematics  
In this note, we show that every graph on n vertices and at most 2n − 4 edges contains a vertex-cut S which is also an independent set of G.  ...  The result is best possible and answers a question proposed by Caro in the a rmative.  ...  A cut S of G which is also an independent set is called an independent cut. A simple connected graph on at least 3 vertices is called a fragile graph if it contains an independent cut.  ... 
doi:10.1016/s0012-365x(01)00226-6 fatcat:rplyxzqqdrdhzfcqprgrsr46ke

Robust-yet-fragile nature of interdependent networks

Fei Tan, Yongxiang Xia, Zhi Wei
2015 Physical Review E  
It is found that interdependent ER random graphs are robust-yet-fragile under both random failures and intentional attack.  ...  Interdependent networks have been shown to be extremely vulnerable based on the percolation model.  ...  Note that the robustness of interdependent random graphs with assortative coupling approaches that of the single ER random graph under random failure.  ... 
doi:10.1103/physreve.91.052809 pmid:26066214 fatcat:5euuwdtvtjcilj2vki7as5ahfe

Decomposing Graphs into a Spanning Tree, an Even Graph, and a Star Forest

Kasper Szabo Lyngsie, Martin Merker
2019 Electronic Journal of Combinatorics  
We prove that every connected graph can be edge-decomposed into a spanning tree, an even graph, and a star forest.  ...  Since G is fragile, the graph G − E(C) is disconnected so there is a vertex u on C which is not in K. Since C is induced, the vertex u has exactly two neighbours on C.  ...  One might be tempted to think that every connected graph admits a decomposition into a spanning tree, an even graph, and a matching.  ... 
doi:10.37236/7970 fatcat:hdwn5e5aqrdwvlc2xnrzcctqim

Sublinear separators, fragility and subexponential expansion [article]

Zdenek Dvorak
2015 arXiv   pre-print
Let G be a subgraph-closed graph class with bounded maximum degree.  ...  This gives a partial converse to a result of Nešetřil and Ossona de Mendez. As an intermediate step, the proof uses a new kind of graph decompositions.  ...  Note that Vs ⊂ Tw ⊂ Be. Let G be a graph and C a class of graphs.  ... 
arXiv:1404.7219v2 fatcat:n3oi4gmvmbdf5gcp3edsthlisa

On Fractional Fragility Rates of Graph Classes

Zdeněk Dvořák, Jean-Sébastien Sereni
2020 Electronic Journal of Combinatorics  
We consider, for every positive integer $a$, probability distributions on subsets of vertices of a graph with the property that every vertex belongs to the random set sampled from this distribution with  ...  of the distribution, the graph $G-X$ has component-size at most $(\Delta(G)-1)^{a+O(\sqrt{a})}$, or treedepth at most $O(a^3\log_2(a))$.  ...  Since the uniform distribution on {X 0 , . . . , X a−1 } is (1/a)-thin, planar graphs are fractionally tw-fragile at rate r(a) = 3a − 3.  ... 
doi:10.37236/8909 fatcat:cmcffslxnndfdene7px3ktey5i

Market Fragility, Systemic Risk, and Ricci Curvature [article]

Romeil Sandhu, Tryphon Georgiou, Allen Tannenbaum
2015 arXiv   pre-print
As a proof of concept, we examine returns from a set of stocks comprising the S\&P 500 over a 15 year span to show that financial crashes are more robust compared to normal "business as usual" fragile  ...  Recent attempts have shown that representing such systems as a weighted graph characterizing the complex web of interacting agents over some information flow (e.g., debt, stock returns, shareholder ownership  ...  (16) We note that the above measure is dependent on a particular graph construction.  ... 
arXiv:1505.05182v1 fatcat:jfoyx25475g2loxrf4lmiavble

PageRank Optimization in Polynomial Time by Stochastic Shortest Path Reformulation [chapter]

Balázs Csanád Csáji, Raphaël M. Jungers, Vincent D. Blondel
2010 Lecture Notes in Computer Science  
The importance of a node in a directed graph can be measured by its PageRank.  ...  We consider the problem of maximizing the PageRank of a node by selecting some of the edges from a set of edges that are under our control.  ...  Note that P is a column (left) stochastic matrix and the chain can be interpreted as an infinite uniform random walk on the graph (e.g., a random surfing).  ... 
doi:10.1007/978-3-642-16108-7_11 fatcat:ylugdked6fb63b2ckl3jr4q2zq

On Resilient Graph Spanners [article]

G. Ausiello, P. G. Franciosa, G. F. Italiano, A. Ribichini
2014 arXiv   pre-print
We introduce and investigate a new notion of resilience in graph spanners. Let S be a spanner of a graph G.  ...  those in the underlying graph G).  ...  Note that, as a special case G \ e denotes the graph obtained after deleting edge e from G.  ... 
arXiv:1303.1559v4 fatcat:gwrgfqmz2zfe5mek4tyankwe24

Ricci curvature: An economic indicator for market fragility and systemic risk

R. S. Sandhu, T. T. Georgiou, A. R. Tannenbaum
2016 Science Advances  
Quantifying the systemic risk and fragility of financial systems is of vital importance in analyzing market efficiency, deciding on portfolio allocation, and containing financial contagions.  ...  At a high level, financial systems may be represented as weighted graphs that characterize the complex web of interacting agents and information flow (for example, debt, stock returns, and shareholder  ...  We note that recent advancements have also explored the concepts of fragility on weighted graphs (and the construction thereof) (16) (17) (18) (19) . In particular, Demetrius et al.  ... 
doi:10.1126/sciadv.1501495 pmid:27386522 pmcid:PMC4928924 fatcat:4noabxuzfrac7nmj37w4jl7pz4

PageRank optimization by edge selection

Balázs Csanád Csáji, Raphaël M. Jungers, Vincent D. Blondel
2014 Discrete Applied Mathematics  
The importance of a node in a directed graph can be measured by its PageRank.  ...  We consider the problem of maximizing the PageRank of a node by selecting some of the edges from a set of edges that are under our control.  ...  Note that P is a column (left) stochastic matrix and the chain can be interpreted as an infinite random walk on the graph (e.g., a random surfing).  ... 
doi:10.1016/j.dam.2014.01.007 fatcat:atamsrcpsna5ljo7uap7elflze

Approximation schemes for bounded distance problems on fractionally treewidth-fragile graphs [article]

Zdeněk Dvořák, Abhiruk Lahiri
2021 arXiv   pre-print
These schemes apply in all fractionally treewidth-fragile graph classes, a property that is true for many natural graph classes with sublinear separators.  ...  We give polynomial-time approximation schemes for monotone maximization problems expressible in terms of distances (up to a fixed upper bound) and efficiently solvable in graphs of bounded treewidth.  ...  Let G be an orientation of a graph G, i.e, uv is an edge of G if and only if the directed graph G contains at least one of the directed edges (u, v) and (v, u); note that we allow G to contain both of  ... 
arXiv:2105.01780v1 fatcat:nwpf5i6rwbe23djuzezmw7fl7e

The Genus of the Erdös-Rényi Random Graph and the Fragile Genus Property

Chris Dowden, Mihyun Kang, Michael Krivelevich, Marc Herbstritt
2018 International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms  
We thus call this the 'fragile genus' property.  ...  on which 'region' m falls into.  ...  of all labelled graphs on [n]).  ... 
doi:10.4230/lipics.aofa.2018.17 dblp:conf/aofa/DowdenKK18 fatcat:xbrzjdm46jc47acnhrswtfbz5y

An Analytical Approach for Insulin-like Growth Factor Receptor 1 and Mammalian Target of Rapamycin Blockades in Ewing Sarcoma [article]

Romeil Sandhu, Salah-Eddine Lamhamedi-Cherradi, Sarah Tannenbaum, Joseph Ludwig, Allen Tannenbaum
2020 arXiv   pre-print
More importantly, given that Ricci curvature is a local property of the system, it is capable of resolving pathway fragility.  ...  In this note, we offer some encouraging yet limited insights in terms of system-level robustness of ES and lay the foundation for scope of future work in which a complete study will be conducted.  ...  Acknowledgements This study was supported by AFOSR grants (FA9550-17-1-0435, FA9550-20-1-0029), NIH grant (R01-AG048769), and a grant from Breast Cancer Research Foundation (grant BCRF-17-193).  ... 
arXiv:1509.03642v2 fatcat:csz6yuufe5fadgbeuofvnszp7i

On classes of graphs with strongly sublinear separators [article]

Zdeněk Dvořák
2018 arXiv   pre-print
A class of graphs has strongly sublinear separators if it is a subclass of G_c,epsilon for some c,epsilon>0.  ...  For real numbers c,epsilon>0, let G_c,epsilon denote the class of graphs G such that each subgraph H of G has a balanced separator of order at most c|V(H)|^1-epsilon.  ...  We say that a class G is fractionally (ε, k)-treewidth-fragile if for each graph G ∈ G there exists an ε-thin probability distribution on coTW k (G).  ... 
arXiv:1710.03117v2 fatcat:g26fyx5gpbfdbectjzge5jssq4

Page 41 of JAMA Internal Medicine Vol. 59, Issue 1 [page]

1937 JAMA Internal Medicine  
in 1928 presented comprehensive mono- graphs on this subject.  ...  Courtial (1705) noted that the bones of aged persons are as fragile as those of persons who have taken large quantities of mercury.  ... 
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