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Contractible cycles in graphs with large minimum degree

1997
*
Discrete Mathematics
*

We show that if k/> 2 and G is a k-connected

doi:10.1016/s0012-365x(96)00017-9
fatcat:ke2sqwlsn5bo7eovy4mqv5dmvu
*graph**with**minimum**degree*at least [3k/2], then, unless 2 ~< k ~< 3 and G is a complete*graph*on k + 2 vertices, G contains an induced*cycle*C such that G - ... Then the resulting*graph*is a (2h + D-connected*graph**with**minimum**degree*3h, but contains no (2h + 1)*contractible**cycle*. Separating triples We start*with*some definitions. Let G be a*graph*. ...*In*this paper, we prove the following refinement of this result (as we shall see*in*Lemma 5.1 , each edge of a k-*contractible*induced*cycle*of a k-connected*graph**with**minimum**degree*at least k + 1 is ...##
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Girth in graphs

1983
*
Journal of combinatorial theory. Series B (Print)
*

Combined

doi:10.1016/0095-8956(83)90067-9
fatcat:373c66tcibeotdsy6bmunetnfy
*with*the above result of Mader [7] , this shows that a*graph*of*minimum**degree*at least 3 and*large*girth can be*contracted*into a*large*complete*graph*. ... Using the results*in*[lo] one can establish a number of results on*cycles**with*many chords*in**graphs*of*large**minimum**degree*. ...##
###
Finding All Global Minimum Cuts In Practice
[article]

2020
*
arXiv
*
pre-print

*In*shared memory we are able to find all

*minimum*cuts of

*graphs*

*with*up to billions of edges and millions of

*minimum*cuts

*in*a few minutes. ... We present a practically efficient algorithm that finds all global

*minimum*cuts

*in*huge undirected

*graphs*. ... For a vertex u ∈ V

*with*

*minimum*vertex

*degree*, the size of the trivial cut ({u}, V \ {u}) is equal to the vertex

*degree*of u. Many algorithms tackling the

*minimum*cut problem use

*graph*

*contraction*. ...

##
###
Finding All Global Minimum Cuts in Practice

2020
*
European Symposium on Algorithms
*

*In*shared memory we are able to find all

*minimum*cuts of

*graphs*

*with*up to billions of edges and millions of

*minimum*cuts

*in*a few minutes. ... We present a practically efficient algorithm that finds all global

*minimum*cuts

*in*huge undirected

*graphs*. ... For a vertex u ∈ V

*with*

*minimum*vertex

*degree*, the size of the trivial cut ({u}, V \ {u}) is equal to the vertex

*degree*of u. Many algorithms tackling the

*minimum*cut problem use

*graph*

*contraction*. ...

##
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Page 2470 of Mathematical Reviews Vol. , Issue 89E
[page]

1989
*
Mathematical Reviews
*

, Roger C. (1-NM)
The

*minimum*number of*cycles**in**graphs**with*given*cycle*rank and small connectivity. ... Let f(k,r) be the*minimum*number of*cycles*possible*in*a k- connected finite simple*graph**with**cycle*rank r. ...##
###
Page 61 of Mathematical Reviews Vol. , Issue 2003A
[page]

2003
*
Mathematical Reviews
*

Since a 2- connected

*graph*of*minimum**degree*at least three and a sufficiently*large*order has a rich*cycle*space, it is suspected that such a*graph*has an (s mod k)-*cycle*for a*large*choice of s and k ... We say that e is*contractible*if F is k-connected. A bowtie is the*graph*consisting of two triangles*with*exactly one vertex*in*common. ...##
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Distribution of contractible edges in k-connected graphs

1990
*
Journal of combinatorial theory. Series B (Print)
*

Our method simultaneously handles the triangle-free case and the case of sufficiently

doi:10.1016/0095-8956(90)90126-k
fatcat:lhiirhqc4zexlgo2tbz3irtrze
*large**minimum**degree*. ... [ 1 ] which states that every vertex of*degree*3*in*a 3-connected*graph*of order at least 5 is incident*with*a 3-*contractible*edge. ... ACKNOWLEDGMENT I am extremely grateful to the referees for their advice including their suggestions which lead to the present short proof of Theorem 8 and a decrease*in*the bound on the*minimum**degree*...##
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Page 2342 of Mathematical Reviews Vol. , Issue 2003d
[page]

2003
*
Mathematical Reviews
*

Summary: “We prove asymptotically tight bounds on the differ- ence between the maximum

*degree*and the*minimum**degree*of a simple*graph**in*terms of its order and of the maximum differ- ence between the ...*In*this work, we mainly show the two following types of conditions*in*order to find generalized p-*cycles**with*maximum connectivity: 1. ...##
###
Practical Fully Dynamic Minimum Cut Algorithms
[article]

2021
*
arXiv
*
pre-print

We present a practically efficient algorithm for maintaining a global

arXiv:2101.05033v1
fatcat:jdup5v7sdnd43cq3dubmjvjnlu
*minimum*cut*in**large*dynamic*graphs*under both edge insertions and deletions. ... The algorithm uses the theoretical foundation and combines it*with*efficient and finely-tuned implementations to give an algorithm that can maintain the global*minimum*cut of a*graph**with*rapid update ...*In*contrast to Family A, most*graphs**in*this*graph*family have a*large*number of*minimum*cuts and generally the*minimum*cut is equal to the*minimum**degree*. ...##
###
On computing the minimum feedback vertex set of a directed graph by contraction operations

2000
*
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
*

This paper is

doi:10.1109/43.833199
fatcat:a3fowx3ffferrmcel3bzrr32jy
*largely*concerned*with*three new and powerful reduction operations. ... Finding the*minimum*feedback vertex set (MFVS)*in*a*graph*is an important problem for a variety of computer-aided design (CAD) applications and*graph*reduction plays an important role*in*solving this intractable ... The reasons are the following:*in*the*graphs**with*lower density, the*contraction*operations can*largely*reduce the sizes of*graphs*and thus result*in*better solutions. ...##
###
Approximating the Minimum Equivalent Digraph

1995
*
SIAM journal on computing (Print)
*

(This result is strengthened slightly

doi:10.1137/s0097539793256685
fatcat:twokz2yxyzhhbev3elyfj7bhci
*in*"On strongly connected digraphs*with*bounded*cycle*length" (1996).) ... This paper gives an approximation algorithm*with*performance guarantee of pi^2/6 ~ 1.64. The algorithm and its analysis are based on the simple idea of*contracting*long*cycles*. ... Starting*with*a*minimum*-size subset that strongly connects the*graph*, repeatedly*contract**cycles**in*the subset until no*cycles*are left. ...##
###
Bounds for the rainbow connection number of graphs

2011
*
Discussiones Mathematicae Graph Theory
*

*In*this paper we show some new bounds for the rainbow connection number of

*graphs*depending on the

*minimum*

*degree*and other

*graph*parameters. Moreover, we discuss sharpness of some of these bounds. ... The rainbow connection number of a connected

*graph*G, denoted rc(G), is the smallest number of colours that are needed

*in*order to make G rainbowconnected. ... The rainbow connection number has been studied for further

*graph*classes

*in*[4] and for

*graphs*

*with*fixed

*minimum*

*degree*

*in*[4, 9, 11] . ...

##
###
A note on minimum degree conditions for supereulerian graphs

2002
*
Discrete Applied Mathematics
*

Furthermore, our main result implies the following strengthening of a theorem of Lai within the class of

doi:10.1016/s0166-218x(01)00278-5
fatcat:ypvvoodybnghrh2dbnjaprexyi
*graphs**with**minimum**degree*¿ 4: If G is a 2-edge-connected*graph*of order n*with*(G) ¿ 4 such that ... Let G be a 2-edge-connected*graph*of order n such that each minimal edge cut S ⊆ E(G)*with*|S| 6 3 satisÿes the property that each component of G − S has order at least (n − 2)=5. ... One might expect that the same conclusions hold without this restriction on the*minimum**degree*. However there exist*graphs**with*a*large**minimum**degree*that are not supereulerian. ...##
###
Algorithm Engineering for Cut Problems
[article]

2021
*
arXiv
*
pre-print

This work develops highly-efficient algorithms for the (global)

arXiv:2108.04566v1
fatcat:4tpyybkhsvg6toiuxkvdvychqu
*minimum*cut problem, the balanced*graph*partitioning problem and the multiterminal cut problem. ...*In*this work, we aim to partition the vertices of a*graph*into multiple blocks while minimizing the number of edges that connect different blocks. ... After every*contraction*, if the*minimum*node*degree**in*the*contracted**graph*is smaller than λ̂, we set λ̂ to the*minimum*node*degree*of the*contracted**graph*. ...##
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Engineering Data Reduction for Nested Dissection
[article]

2020
*
arXiv
*
pre-print

*In*this paper, we engineer new data reduction rules for the

*minimum*fill-

*in*problem, which significantly reduce the size of the

*graph*while producing an equivalent (or near-equivalent) instance. ... Many applications rely on time-intensive matrix operations, such as factorization, which can be sped up significantly for

*large*sparse matrices by interpreting the matrix as a sparse

*graph*and computing ... Let C = (V, E) be a

*cycle*

*with*|V | ≥ 3 nodes. Any ordering of C is a

*minimum*fill-

*in*ordering. ...

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