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## Filters

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Making the components of a graph k-connected

2007
*
Discrete Applied Mathematics
*

For every integer

doi:10.1016/j.dam.2006.07.007
fatcat:ow6lukeq4rdmjavlrohr4tnu4m
*k*2 and*graph*G, consider*the*following natural procedure: if G has*a**component*G that is not*k*-*connected*, remove G if |G |*k*, otherwise remove*a*cutset U ⊂ V (G ) with |U | <*k*; do*the*... same with*the*remaining*graph*until only*k*-*connected**components*are left or all vertices are removed. ... Given an integer*k*2 and*a**graph*G,*a**k*-cutting procedure*of*G is defined as follows: while G has*a**component*G that is not*k*-*connected*and V (G) = ∅ do if G has*a**component*G with |G |*k*then set G = ...##
###
Making Bidirected Graphs Strongly Connected
[article]

2017
*
arXiv
*
pre-print

We consider problems to

arXiv:1709.00824v1
fatcat:fwtxgtmopje3fhsa4ahmvtf36u
*make**a*given bidirected*graph*strongly*connected*with minimum cardinality*of*additional signs or additional arcs. ... For*the*former problem, we show*the*minimum number*of*additional signs and give*a*linear-time algorithm for finding an optimal solution. ... Related Works It is obvious that*the*minimum number*of*additional edges to*make**a*given undirected*graph**connected*is fewer than*the*number*of**connected**components**of**a*given*graph*by one. ...##
###
Conflict-free connection numbers of line graphs
[article]

2017
*
arXiv
*
pre-print

For

arXiv:1705.05317v1
fatcat:kilqoblqvve5jlcn7cl6plitli
*a**connected**graph*G,*the*conflict-free*connection*number*of*G, denoted by cfc(G), is defined as*the*minimum number*of*colors that are required to*make*G conflict-free*connected*. ... Secondly, we get*the*exact value*of**the*conflict-free*connection*number*of**a**connected*claw-free*graph*, especially*a**connected*line*graph*. ... For*a**connected**graph*G,*the*rainbow*connection*number*of*G, denoted by rc(G), is defined as*the*minimum number*of*colors that are needed to*make*G rainbow*connected*. ...##
###
On the maximum value of conflict-free verex-connection number of graphs
[article]

2017
*
arXiv
*
pre-print

*A*vertex-colored

*graph*is said to be conflict-free vertex-

*connected*if any two vertices

*of*

*the*

*graph*are

*connected*by

*a*conflict-free path. ...

*The*conflict-free vertex-

*connection*number, denoted by vcfc(G), is defined as

*the*smallest number

*of*colors required to

*make*G conflict-free vertex-

*connected*. ...

*The*conflict-free

*connection*number

*of*

*a*

*connected*

*graph*, denoted by cf c(G), is defined

*a*

*the*smallest number

*of*colors required to

*make*G conflict-free

*connected*. ...

##
###
Connectivity structure of bipartite graphs via the KNC-plot

2008
*
Proceedings of the international conference on Web search and web data mining - WSDM '08
*

*The*KNC-plot shows

*the*degradation

*of*

*connectivity*

*of*

*the*

*graph*as

*a*function

*of*

*k*. ... In this paper we introduce

*the*

*k*-neighbor

*connectivity*plot, or KNC-plot, as

*a*tool to study

*the*macroscopic

*connectivity*structure

*of*sparse bipartite

*graphs*. ... We extract

*the*size

*of*

*the*largest

*component*and

*the*number

*of*

*connected*

*components*in

*the*

*graph*G

*k*, as

*a*function

*of*

*k*, and plots

*the*results in

*the*KNC-plot. ...

##
###
Minimum 2-vertex-twinless connected spanning subgraph problem
[article]

2020
*
arXiv
*
pre-print

Given

arXiv:2001.03788v1
fatcat:dfud4aze7zempdiaxd5bdn23tu
*a*2-vertex-twinless*connected*directed*graph*G=(V,E),*the*minimum 2-vertex-twinless*connected*spanning subgraph problem is to find*a*minimum cardinality edge subset E^t⊆ E such that*the*subgraph ... Let G^1 be*a*minimal 2-vertex-*connected*subgraph*of*G. ...*The*results*of*[2, 10] imply that every minimal 2-vertex-*connected*spanning subgraph has at most 4n edges. ...##
###
Local Reconstructors and Tolerant Testers for Connectivity and Diameter
[chapter]

2013
*
Lecture Notes in Computer Science
*

For this model, we achieve local property reconstructors for

doi:10.1007/978-3-642-40328-6_29
fatcat:sw7xhprozbchfjmxsbeonjzsq4
*the*properties*of**connectivity*and*k*-*connectivity*in undirected*graphs*, and*the*property*of*strong*connectivity*in directed*graphs*. ... matrix*of**a*"correction"*of**the**graph*, i.e.*a**graph*which has*the*property and is close to*the*given*graph*. ... Ideally, we have exactly one special vertex in each*connected**component*, since this number*of*edges is both necessary and sufficient to*make**the**graph**connected*. ...##
###
Local reconstructors and tolerant testers for connectivity and diameter
[article]

2013
*
arXiv
*
pre-print

For this model, we achieve local property reconstructors for

arXiv:1208.2956v2
fatcat:kdwhlqywandqfp3q2ifjrtwjcm
*the*properties*of**connectivity*and*k*-*connectivity*in undirected*graphs*, and*the*property*of*strong*connectivity*in directed*graphs*. ... matrix*of**a*"correction"*of**the**graph*, i.e.*a**graph*which has*the*property and is close to*the*given*graph*. ... Ideally, we have exactly one special vertex in each*connected**component*, since this number*of*edges is both necessary and sufficient to*make**the**graph**connected*. ...##
###
One-Way Trail Orientations
[article]

2017
*
arXiv
*
pre-print

Monthly, 1980] for mixed multigraphs states that

arXiv:1708.07389v1
fatcat:k43or44uczh6jc2ifidonwmywu
*the*undirected edges*of**a*mixed multigraph can be oriented*making**the*resulting directed*graph*strongly*connected*exactly when*the*mixed*graph*is*connected*... Given*a**graph*, does there exist an orientation*of**the*edges such that*the*resulting directed*graph*is strongly*connected*? Robbins' theorem [Robbins, Am. Math. ... Let*a*1 , . . . ,*a**k*be*the*number*of*vertices in*the*"large" 3-edge*connected**components*. ...##
###
Page 6543 of Mathematical Reviews Vol. , Issue 93m
[page]

1993
*
Mathematical Reviews
*

This

*makes**the*operation applica- ble to hypergraphs and*makes*possible*a*recursive definition*of**k*-*connected**components*(which obviously are not*the*traditional ones in relation to*the*original*graph*) ...*The*author does not*make*contact with Matula’s concepts*of**k*-*components*and*k*-blocks and good algorithms for their detection, which are nearer to*the*traditional theory [see, e.g., D. W. ...##
###
Tight query complexity bounds for learning graph partitions
[article]

2022
*
arXiv
*
pre-print

Given

arXiv:2112.07897v2
fatcat:ejm7h5fasbhlpexubegp7hctmm
*a*partition*of**a**graph*into*connected**components*,*the*membership oracle asserts whether any two vertices*of**the**graph*lie in*the*same*component*or not. ... We prove that for n≥*k*≥ 2, learning*the**components**of*an n-vertex hidden*graph*with*k**components*requires at least (*k*-1)n- k2 membership queries. ... Research*of**the*first author was supported by ERC Advanced Grant 101020255. ...##
###
Chromatic classes of 2-connected (n,n+4)-graphs with at least four triangles

2004
*
Discrete Mathematics
*

Notation and basic results

doi:10.1016/j.disc.2003.05.003
fatcat:nwh7iqm4d5gy3dx56f7cjqkq5q
*The*following are some useful known results and techniques for determining*the*chromatic polynomial*of**a**graph*. ... In this paper, we shall determine all*the*chromatic equivalence classes*of*2-*connected*(n, n + 4)*graphs*with at least four triangles. ... (iv)*The**graph*J 18 is*a*disconnected (n, n + 2)-*graph*having four*components*. If three*of**the**components*are*connected*by two paths, we get*a*disconnected (n, n + 4)-*graph*. ...##
###
The Topology of Competitively Constructed Graphs

2014
*
Electronic Journal of Combinatorics
*

We show

doi:10.37236/3942
fatcat:dtvtjc2bnzdjfldlqifzczb7ey
*a*sharp topological threshold for this game: for*the*case $*k*=3$*a*player can ensure*the*resulting*graph*is planar, while for*the*case $*k*=4$,*a*player can force*the*appearance*of*arbitrarily large ... We consider*a*simple game,*the*$*k*$-regular*graph*game, in which players take turns adding edges to an initially empty*graph*subject to*the*constraint that*the*degrees*of*vertices cannot exceed $*k*$. ... Thus there is no surface S for which*a*player*of**the*4-regular*graph*game can ensure that*the**connected**components**of**the*result has*a*drawing on S. ...##
###
New Clusterization Method Based on Graph Connectivity Search

2017
*
Journal of Siberian Federal University Mathematics & Physics
*

*The*method is based on

*a*sequential elimination

*of*

*the*longest distances in dataset, so that

*the*relevant

*graph*looses some edges.

*The*method stops when

*the*

*graph*becomes disconnected. ... Hence, an examination for

*graph*

*connectivity*must be changed for

*the*

*connected*

*components*search, in

*the*

*graph*. There is

*a*number

*of*algorithms to seek for

*connected*

*components*, in

*a*

*graph*. ... Indeed, any

*graph*is unambiguously decomposed into

*a*number

*of*

*connected*

*components*;

*a*

*connected*

*graph*has

*the*unique

*connected*

*component*, and vice versa [8] [9] [10] . ...

##
###
Conflict-free connection number and independence number of a graph
[article]

2020
*
arXiv
*
pre-print

*The*conflict-free

*connection*number

*of*

*a*

*connected*

*graph*G, denoted by cfc(G), is defined as

*the*minimum number

*of*colors that are required in order to

*make*G conflict-free

*connected*. ... In this paper, we investigate

*the*relation between

*the*conflict-free

*connection*number and

*the*independence number

*of*

*a*

*graph*. ... Acknowledgement 1 This work was done during

*the*first author was visiting Nankai University. ...

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