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A note on fragile graphs

2002
*
Discrete Mathematics
*

In this

doi:10.1016/s0012-365x(01)00226-6
fatcat:rplyxzqqdrdhzfcqprgrsr46ke
*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. ...##
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Robust-yet-fragile nature of interdependent networks

2015
*
Physical Review E
*

It is found that interdependent ER random

doi:10.1103/physreve.91.052809
pmid:26066214
fatcat:5euuwdtvtjcilj2vki7as5ahfe
*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. ...##
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Decomposing Graphs into a Spanning Tree, an Even Graph, and a Star Forest

2019
*
Electronic Journal of Combinatorics
*

We prove that every connected

doi:10.37236/7970
fatcat:hdwn5e5aqrdwvlc2xnrzcctqim
*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. ...##
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Sublinear separators, fragility and subexponential expansion
[article]

2015
*
arXiv
*
pre-print

Let G be

arXiv:1404.7219v2
fatcat:n3oi4gmvmbdf5gcp3edsthlisa
*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*. ...##
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On Fractional Fragility Rates of Graph Classes

2020
*
Electronic Journal of Combinatorics
*

We consider, for every positive integer $

doi:10.37236/8909
fatcat:cmcffslxnndfdene7px3ktey5i
*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. ...##
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Market Fragility, Systemic Risk, and Ricci Curvature
[article]

2015
*
arXiv
*
pre-print

As

arXiv:1505.05182v1
fatcat:jfoyx25475g2loxrf4lmiavble
*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. ...##
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PageRank Optimization in Polynomial Time by Stochastic Shortest Path Reformulation
[chapter]

2010
*
Lecture Notes in Computer Science
*

The importance of

doi:10.1007/978-3-642-16108-7_11
fatcat:ylugdked6fb63b2ckl3jr4q2zq
*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). ...##
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On Resilient Graph Spanners
[article]

2014
*
arXiv
*
pre-print

We introduce and investigate

arXiv:1303.1559v4
fatcat:gwrgfqmz2zfe5mek4tyankwe24
*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. ...##
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Ricci curvature: An economic indicator for market fragility and systemic risk

2016
*
Science Advances
*

Quantifying the systemic risk and

doi:10.1126/sciadv.1501495
pmid:27386522
pmcid:PMC4928924
fatcat:4noabxuzfrac7nmj37w4jl7pz4
*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. ...##
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PageRank optimization by edge selection

2014
*
Discrete Applied Mathematics
*

The importance of

doi:10.1016/j.dam.2014.01.007
fatcat:atamsrcpsna5ljo7uap7elflze
*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). ...##
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Approximation schemes for bounded distance problems on fractionally treewidth-fragile graphs
[article]

2021
*
arXiv
*
pre-print

These schemes apply in all fractionally treewidth-

arXiv:2105.01780v1
fatcat:nwpf5i6rwbe23djuzezmw7fl7e
*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 ...##
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The Genus of the Erdös-Rényi Random Graph and the Fragile Genus Property

2018
*
International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms
*

We thus call this the '

doi:10.4230/lipics.aofa.2018.17
dblp:conf/aofa/DowdenKK18
fatcat:xbrzjdm46jc47acnhrswtfbz5y
*fragile*genus' property. ...*on*which 'region' m falls into. ... of all labelled*graphs**on*[n]). ...##
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An Analytical Approach for Insulin-like Growth Factor Receptor 1 and Mammalian Target of Rapamycin Blockades in Ewing Sarcoma
[article]

2020
*
arXiv
*
pre-print

More importantly, given that Ricci curvature is

arXiv:1509.03642v2
fatcat:csz6yuufe5fadgbeuofvnszp7i
*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). ...##
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On classes of graphs with strongly sublinear separators
[article]

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). ...

##
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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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