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The complexity of graph connectivity
[chapter]

1992
*
Lecture Notes in Computer Science
*

There seem to be two complimentary reasons for this wide interest in

doi:10.1007/3-540-55808-x_10
fatcat:5cjmn5ndo5eg7jyfcyump25mry
*the**complexity**of**graph**connectivity*. ... In this paper we survey*the*major developments in understanding*the**complexity**of**the**graph**connectivity*problem in several computational models, and highlight some challenging open problems. ... Acknowledgments I wish to thank Moni Naor, Ilan Newman and Noam Nisan for reading and improving an earlier version*of*this paper. ...##
###
The Parameterised Complexity of Counting Connected Subgraphs and Graph Motifs
[article]

2014
*
arXiv
*
pre-print

We show that exactly counting

arXiv:1308.1575v3
fatcat:fryb77k3gze65clacn6kgap4pq
*the*number*of**connected*induced k-vertex subgraphs in an n-vertex*graph*is #W[1]-hard, but on*the*other hand there exists an FPTRAS for*the*problem; more generally, we show ... We then apply these results to a counting version*of**the**Graph*Motif problem. ... ÓÒÒ Ø ÁÒ Ù ËÙ Ö Ô*The*number*of*multicoloured*connected*induced subgraphs in a*graph*G can be computed by inclusionexclusion from*the*numbers*of**connected*induced subgraphs in*the*2 k subgraphs*of*G induced ...##
###
The complexity of optimal design of temporally connected graphs
[article]

2016
*
arXiv
*
pre-print

We study

arXiv:1502.04579v3
fatcat:jw5d3xrqwbbmhi337gft5ye2wm
*the*design*of*small cost temporally*connected**graphs*, under various constraints. ... We then consider*the*case in which a designer*of*temporal*graphs*can freely choose availability instances for all edges and aims for temporal*connectivity*with very small cost;*the*cost is*the*total number ... We resolve here*the**complexity**of*finding*the*maximum number*of*redundant labels in any given temporal*graph*. ...##
###
The Exponential Time Complexity of Computing the Probability That a Graph Is Connected
[chapter]

2010
*
Lecture Notes in Computer Science
*

We show that for every probability p with 0 < p < 1, computation

doi:10.1007/978-3-642-17493-3_19
fatcat:mkayelond5ekfdgsf3kz6rdxze
*of*all-terminal*graph*reliability with edge failure probability p requires time exponential in Omega(m/ log^2 m) for simple*graphs**of*m ... edges under*the*Exponential Time Hypothesis. ... Since n ≤ m for*connected**graphs*,*the*result implies*the*lower bound exp(Ω(n/ log 2 n)) in terms*of**the*parameter n,*the*number*of*vertices*of**the*input*graph*. ...##
###
On the complexity of computing the k-restricted edge-connectivity of a graph
[article]

2016
*
arXiv
*
pre-print

Very recently, in

arXiv:1502.07659v2
fatcat:jbak456vqbhrno54sb7ypns7am
*the*parameterized*complexity*community*the*notion*of*good edge separation*of*a*graph*has been defined, which happens to be essentially*the*same as*the*k-restricted edge-*connectivity*. ...*The*k-restricted edge-*connectivity**of*a*graph*G, denoted by λ_k(G), is defined as*the*minimum size*of*an edge set whose removal leaves exactly two*connected*components each containing at least k vertices ... We would like to thank*the*anonymous referees for helpful remarks that improved*the*presentation*of**the*manuscript. ...##
###
On Connectivity of the Facet Graphs of Simplicial Complexes
[article]

2015
*
arXiv
*
pre-print

*The*paper studies

*the*

*connectivity*properties

*of*facet

*graphs*

*of*simplicial

*complexes*

*of*combinatorial interest. ... It is also shown that

*the*facet

*graph*

*of*a d-cycle cannot be split into more than s

*connected*components by removing at most s vertices. ... Theorem 4.1 has an immediate implication on

*connectivity*

*of*

*the*facet

*graphs*

*of*d-biconnected sets S

*of*d-

*complexes*. ...

##
###
On the complexity of structure and substructure connectivity of graphs
[article]

2021
*
arXiv
*
pre-print

In this paper, we characterize

arXiv:2110.05917v1
fatcat:rsnsdksojfgf7kli55n7mewube
*the**complexity**of*determining structure and substructure*connectivity**of**graphs*, showing that they are both NP-complete. ...*The**connectivity**of*a*graph*is an important parameter to measure its reliability. Structure and substructure*connectivity*are two novel generalizations*of**the**connectivity*. ... This completes*the*proof. Remark.*The*authors [2] constructed*the*similar*graph*G ′ as in Theorem 2 to prove NP-completeness*of*neighbor*connectivity*by reducing from dominating set problem. ...##
###
On the Eccentric Connectivity Polynomial of ℱ-Sum of Connected Graphs

2020
*
Complexity
*

*The*eccentric

*connectivity*polynomial (ECP)

*of*a

*connected*

*graph*G=VG,EG is described as ξcG,y=∑a∈VGdegGayecGa, where ecGa and degGa represent

*the*eccentricity and

*the*degree

*of*

*the*vertex a, respectively ... In this article, we work out

*the*ECP

*of*ℱ-sum

*of*

*graphs*. ... Acknowledgments is research was supported by UPAR Grant

*of*United Arab Emirates University (UAEU), Al Ain, UAE, via Grant nos. G00002590 and G00003271. ...

##
###
The complexity of determining the rainbow vertex-connection of graphs
[article]

2011
*
arXiv
*
pre-print

In this paper, we study

arXiv:1101.3126v1
fatcat:5yqx5shnp5gidfsfx3wftbgop4
*the*computational*complexity**of*vertex-rainbow*connection**of**graphs*and prove that computing rvc(G) is NP-Hard. ...*The*rainbow vertex-*connection**of*a*connected**graph*G, denoted by rvc(G), is*the*smallest number*of*colors that are needed in order to make G rainbow vertex-*connected*. ... Motivated by*the*proof*of*[3] , we consider*the*computational*complexity**of*rainbow vertex-*connection*rvc(G)*of**graphs*. ...##
###
The connectivity of graphs of graphs with self-loops and a given degree sequence

2018
*
Journal of Complex Networks
*

*The*same classification scheme to characterize degree sequences can be used to prove that, for all degree sequences, loopy

*graphs*are

*connected*by a combination

*of*double and triple edge swaps. ... However, while double edge-swaps can transform, for any fixed degree sequence, any two

*graphs*inside

*the*classes

*of*simple

*graphs*, multigraphs, and pseudographs, this is not true for

*graphs*which allow ... Categorizing

*the*components

*of*G For any

*graph*G i ∈ G, let V(G i ) be

*the*

*graphs*

*connected*to G i with

*the*maximum number

*of*self-loops. ...

##
###
The Complexity of Connectivity Problems in Forbidden-Transition Graphs and Edge-Colored Graphs
[article]

2020
*
arXiv
*
pre-print

We initiate

arXiv:2009.12892v1
fatcat:cfdsfxzgtfhjtofbpuwljffaoe
*the*study*of*fundamental*connectivity*problems from*the*point*of*view*of*parameterized*complexity*, including an in-depth study*of*tractability with regards to various*graph*-width parameters ...*The*notion*of*forbidden-transition*graphs*allows for a robust generalization*of*walks in*graphs*. ... As*the*notion*of*properly colored walks in edge-colored*graphs*generalizes walks in directed*graphs*,*the*problem in question is more general than finding a cycle*of*length at least k in a directed*graph*...##
###
The Complexity of Optimal Design of Temporally Connected Graphs

2017
*
Theory of Computing Systems
*

We study

doi:10.1007/s00224-017-9757-x
pmid:32025196
pmcid:PMC6979514
fatcat:yinjlbjl45e4jfjyezykhsupci
*the*design*of*small cost temporally*connected**graphs*, under various constraints. ... We then consider*the*case in which a designer*of*temporal*graphs*can freely choose availability instances for all edges and aims for temporal*connectivity*with very small cost;*the*cost is*the*total number ... We wish to give special thanks to*the*reviewers for*the*suggestion*of**the*theoretical proof*of*Theorem 4(a), which now replaces*the*program code previously used in*the*proof. ...##
###
On the complexity of partitioning graphs into connected subgraphs

1985
*
Discrete Applied Mathematics
*

This paper is mainly concerned with

doi:10.1016/0166-218x(85)90008-3
fatcat:l5hj2nqqvngqdecuphgxe6giem
*the*computational*complexity**of*determining whether or not*the*vertices*of*a*graph*can be partitioned into equal sized subsets so that each subset induces a particular ... type*of**graph*. ... (b) If G is 4-edge*connected*, then G has a*connected*k-partition for aN k.*The*question*of**the**complexity**of*Ek(*connected*) was left open in*the*above paper. ...##
###
On the connectivity of infinite graphs and 2-complexes

1999
*
Discrete Mathematics
*

This paper contains a study

doi:10.1016/s0012-365x(98)00033-8
fatcat:htchfbmxv5bhtcvrhqz7nbzb74
*of**the**connectivity**of*infinite*graphs*and 2-*complexes*. Various*connectivity*types are defined and relationships among them are given. ... In addition new Menger-Whitney type theorems are stated for both*graphs*and 2-*complexes*. ... Acknowledgements This work was partially supported by*the*project DGICYT PB96-1374. ...##
###
A short proof of a conjecture on the higher connectivity of graph coloring complexes
[article]

2005
*
arXiv
*
pre-print

*The*Hom-

*complexes*were introduced by Lovasz to study topological obstructions to

*graph*colorings. ... It was conjectured by Babson and Kozlov, and proved by Cukic and Kozlov, that Hom(G,K_n) is (n-d-2)-

*connected*, where d is

*the*maximal degree

*of*a vertex

*of*G. We give a short proof

*of*

*the*conjecture. ... A regular cell

*complex*∆ is m-

*connected*if there is a family

*of*subcomplexes {∆ i } such that ∆ = ∪∆ i , all

*of*

*the*subcomplexes ∆ i are m-

*connected*, and all

*of*

*the*intersection

*of*several ∆ i 's are ( ...

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