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Contractible cycles in graphs with large minimum degree

Yoshimi Egawa
1997 Discrete Mathematics  
We show that if k/> 2 and G is a k-connected 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  ... 
doi:10.1016/s0012-365x(96)00017-9 fatcat:ke2sqwlsn5bo7eovy4mqv5dmvu

Girth in graphs

Carsten Thomassen
1983 Journal of combinatorial theory. Series B (Print)  
Combined 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.  ... 
doi:10.1016/0095-8956(83)90067-9 fatcat:373c66tcibeotdsy6bmunetnfy

Finding All Global Minimum Cuts In Practice [article]

Monika Henzinger, Alexander Noe, Christian Schulz, Darren Strash
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.  ... 
arXiv:2002.06948v1 fatcat:t4v3g2h2yfgennbymgyfyfitmq

Finding All Global Minimum Cuts in Practice

Monika Henzinger, Alexander Noe, Christian Schulz, Darren Strash, Peter Sanders, Fabrizio Grandoni, Grzegorz Herman
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.  ... 
doi:10.4230/lipics.esa.2020.59 dblp:conf/esa/HenzingerN0S20 fatcat:mykbuoxpzzhdxfnj57ababklkq

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.  ... 

Distribution of contractible edges in k-connected graphs

Nathaniel Dean
1990 Journal of combinatorial theory. Series B (Print)  
Our method simultaneously handles the triangle-free case and the case of sufficiently 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  ... 
doi:10.1016/0095-8956(90)90126-k fatcat:lhiirhqc4zexlgo2tbz3irtrze

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]

Monika Henzinger, Alexander Noe, Christian Schulz
2021 arXiv   pre-print
We present a practically efficient algorithm for maintaining a global 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.  ... 
arXiv:2101.05033v1 fatcat:jdup5v7sdnd43cq3dubmjvjnlu

On computing the minimum feedback vertex set of a directed graph by contraction operations

Hen-Ming Lin, Jing-Yang Jou
2000 IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems  
This paper is 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.  ... 
doi:10.1109/43.833199 fatcat:a3fowx3ffferrmcel3bzrr32jy

Approximating the Minimum Equivalent Digraph

Samir Khuller, Balaji Raghavachari, Neal Young
1995 SIAM journal on computing (Print)  
(This result is strengthened slightly 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.  ... 
doi:10.1137/s0097539793256685 fatcat:twokz2yxyzhhbev3elyfj7bhci

Bounds for the rainbow connection number of graphs

Ingo Schiermeyer
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] .  ... 
doi:10.7151/dmgt.1553 fatcat:r67ruxle5veoxagwomnjnsew2i

A note on minimum degree conditions for supereulerian graphs

H.J. Broersma, Liming Xiong
2002 Discrete Applied Mathematics  
Furthermore, our main result implies the following strengthening of a theorem of Lai within the class of 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.  ... 
doi:10.1016/s0166-218x(01)00278-5 fatcat:ypvvoodybnghrh2dbnjaprexyi

Algorithm Engineering for Cut Problems [article]

Alexander Noe
2021 arXiv   pre-print
This work develops highly-efficient algorithms for the (global) 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.  ... 
arXiv:2108.04566v1 fatcat:4tpyybkhsvg6toiuxkvdvychqu

Engineering Data Reduction for Nested Dissection [article]

Wolfgang Ost, Christian Schulz, Darren Strash
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.  ... 
arXiv:2004.11315v1 fatcat:ceirz26xubguff77jzpna52c4a
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