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Removable Edges in Longest Cycles of 4-Connected Graphs

2004
*
Graphs and Combinatorics
*

Let G be a

doi:10.1007/s00373-004-0566-z
fatcat:odgvvfolnbbv3oz4tb3ilbxi7e
*4*-*connected**graph*. ... The authors are greatly indebted to a referee for his valuable suggestions and comments, which are very helpful to improve the proof*of*our main result Lemma 3.3. ...*In*this paper we shall focus on the study*of**removable**edges**in**4*-*connected**graphs*. First*of*all, we give the definition*of*a*removable**edge*for a*4*-*connected**graph*. ...##
###
An approximation algorithm for the longest cycle problem in solid grid graphs
[article]

2015
*
arXiv
*
pre-print

More precisely, our algorithm finds a

arXiv:1502.07085v1
fatcat:joickxegzvervhorobypfuvxcy
*cycle**of*length at least 2n/3+1*in*2-*connected*n-node solid grid*graphs*. Keywords:*Longest**cycle*, Hamiltonian*cycle*, Approximation algorithm, Solid grid*graph*. ... Although, the Hamiltonicity*of*solid grid*graphs*are polynomial-time decidable, the complexity*of*the*longest**cycle*problem*in*these*graphs*is still open. ... There are few classes*of**graphs**in*which the*longest*path or the*longest**cycle*problems are polynomial [5, 11, 12, 15, 16, 17] .*In*the case*of*grid*graphs*, Itai et al. ...##
###
Exact Algorithms for Finding Longest Cycles in Claw-Free Graphs

2011
*
Algorithmica
*

For a claw-free

doi:10.1007/s00453-011-9576-4
fatcat:4amjcwpgzjh3dlcntei2lxslc4
*graph*G, finding a*longest**cycle*is equivalent to finding a closed trail (i.e., a*connected*even subgraph, possibly consisting*of*a single vertex) that dominates the largest number*of**edges*... The*Longest**Cycle*problem,*in*which the task is to find a*cycle**of*maximum length, is a natural generalization*of*the Hamiltonian*Cycle*problem. ... We thank an anonymous referee for some useful comments that helped us to improve the readability*of*our paper. ...##
###
Page 1389 of Mathematical Reviews Vol. , Issue 90C
[page]

1990
*
Mathematical Reviews
*

L. (

*4*-LNDIC)*Removing*monotone*cycles*from orientations.*Graph*theory*in*memory*of*G. A. Dirac (Sandbjerg, 1985), 355-362, Ann. Discrete Math., 41, North-Holland, Amsterdam-New York, 1989. ... [Jackson, Bill] (*4*-LNDG) A note concerning some conjectures on cyclically*4*-*edge**connected*3-regular*graphs*.*Graph*theory*in*memory*of*G. A. Dirac (Sandbjerg, 1985), 171-177, Ann. ...##
###
Approximation algorithms for minimum tree partition

1998
*
Discrete Applied Mathematics
*

It runs

doi:10.1016/s0166-218x(98)00052-3
fatcat:pzs7l6hsj5cfjleoy76cfd762m
*in*0( ~'*4*~ + n') time (n = 1 VI) and comes within a factor*of*2p -1*of*optimal. When the sets' sizes are all equal this algorithm runs*in*O(n*) time. ...*In*this problem, it is required to partition the set*of*n customers into subsets minimizing the length*of*nets required to*connect*all the customers to the communication centers. ... (e-cou for*edge*-count). while (done = 0)*Remove*the e-cou*longest**edges**in*ET. A set*of**connected*components {Ti}~~~+' is created (T; is a spunning tree*of*VT,). ...##
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On the Intersections of Longest Cycles in a Graph

1995
*
Experimental Mathematics
*

The circumference

doi:10.1080/10586458.1995.10504306
fatcat:o4osopflenfizmitzbbvtvbaxe
*of*G is the length*of*a*longest**cycle**in*G. A Hamiltonian*cycle**in*G is a*cycle*passing through every vertex*of*V . A*graph*is*connected*if any two vertices are joined by a path. ... As remarked*in*Gr otschel 1984] , a considerable amount*of*work has been done regarding the length*of**longest**cycles**in*various*graphs*, but not much attention has been paid to how these*longest**cycles*...##
###
My Research Visiting Card in Hamiltonian Graph Theory
[article]

2012
*
arXiv
*
pre-print

*In*Theorems 1-3 we give three lower bounds for the length

*of*a

*longest*

*cycle*C

*of*a

*graph*G

*in*terms

*of*minimum degree δ,

*connectivity*κ and parameters p̅, c̅ - the lengths

*of*a

*longest*path and

*longest*... We present eighteen exact analogs

*of*six well-known fundamental Theorems (due to Dirac, Nash-Williams and Jung)

*in*hamiltonian

*graph*theory providing alternative compositions

*of*

*graph*invariants. ... Theorem 10 [9] (2011) (Analog

*of*Theorem D) If G is a

*4*-

*connected*

*graph*then either c ≥

*4*δ − κ −

*4*or each

*longest*

*cycle*

*in*G is a dominating

*cycle*.

*longest*

*cycle*

*in*G is a dominating

*cycle*. ...

##
###
Cubic graphs with large circumference deficit
[article]

2013
*
arXiv
*
pre-print

*In*contrast, the dominating

*cycle*conjecture implies that the circumference ratio

*of*a cyclically

*4*-

*edge*-

*connected*cubic

*graph*is at least 0.75. ... The circumference c(G)

*of*a

*graph*G is the length

*of*a

*longest*

*cycle*. ... This work was supported from the APVV grants APVV-0223-10 and ESF-EC-0009-10 within the EUROCORES Programme EUROGIGA (project GRe-GAS)

*of*the European Science Foundation. ...

##
###
On longest cycles in essentially 4-connected planar graphs

2016
*
Discussiones Mathematicae Graph Theory
*

Moreover, new lower bounds on the length

doi:10.7151/dmgt.1875
fatcat:onluwzbq3zb3ze4iacrfvuvmxe
*of*a*longest**cycle**of*G are presented if G is an essentially*4*-*connected*planar*graph**of*maximum degree*4*or G is an essentially*4*-*connected*maximal planar*graph*... A planar 3-*connected**graph*G is essentially*4*-*connected*if, for any 3separator S*of*G, one component*of*the*graph*obtained from G by*removing*S is a single vertex. ...*In*the present paper we are interested*in*the length*of**longest**cycles**of*an essentially*4*-*connected*planar*graph*. ...##
###
Page 2998 of Mathematical Reviews Vol. , Issue 92f
[page]

1992
*
Mathematical Reviews
*

(D-HANN) About the number

*of**removable**edges**in*n-regular n-*edge*-*connected**graphs*. Contemporary methods*in**graph*theory, 405-408, Bibliographisches Inst., Mannheim, 1990. ... An*edge*f*of*such a*graph*G is called*removable*if every two vertices*of*G not incident to f are*connected*by n pairwise*edge*-disjoint paths*in*G— f. ...##
###
Encryption and Decryption Process Using Edge Magic Labeling

2019
*
Journal of Physics, Conference Series
*

*In*this paper we are using

*edge*magic labeling

*graph*to encrypt a message. Then using

*Longest*

*cycle*path technique we decrypt the message by inverse matrix multiplication. ... adjacency matrix F1 x Find the

*longest*

*cycle*path

*in*the plain

*graph*and

*remove*all other

*edges*

*in*it x Using this

*cycle*

*graph*find the adjacency matrix F2 x Modify the F2 with the order given

*in*D ... This type

*of*labeled

*graph*is also called a weighted

*graph*.

*In*this work we define a problem

*of*

*longest*closed path which forms a

*cycle*

*graph*. ...

##
###
Page 6434 of Mathematical Reviews Vol. , Issue 95k
[page]

1995
*
Mathematical Reviews
*

Summary: “The

*edge*-neighbor-*connectivity**of*a*graph*G is the minimum size*of*all*edge*-cut-strategies*of*G, where an*edge*-cut- strategy consists*of*a set*of*common*edges**of*double stars whose*removal*disconnects ... The authors show that the length*of*a*longest**cycle**in*a threshold*graph*equals the size*of*a largest matching*in*a certain highly structured bipartite*graph*(so highly structured,*in*fact, that a matching ...##
###
Page 5751 of Mathematical Reviews Vol. , Issue 2003h
[page]

2003
*
Mathematical Reviews
*

*In*this paper, we study the problem

*of*2003h:05125 disconnecting p-

*cycles*by

*removing*nontrivial subsets

*of*vertices or

*edges*. ... Bondy [“

*Longest*paths and

*cycles*

*in*

*graphs*

*of*high degree”, Res. Rep. CORR 80-16, Univ. ...

##
###
Page 1398 of Mathematical Reviews Vol. , Issue 86d
[page]

1986
*
Mathematical Reviews
*

Fleischner): Every 3-regular cyclically

*4*-*edge**connected**graph*G has a*cycle*C such that each*edge**of*G is incident with a vertex*of*C. ...*of**cycles*whose*edge*-deletion preserves 2-*connectivity**in**graphs*. ...##
###
Destroying longest cycles in graphs and digraphs

2015
*
Discrete Applied Mathematics
*

*graph*one can destroy all the

*longest*

*cycles*by deleting at most one third

*of*the vertices. ... The Petersen

*graph*demonstrates that this result cannot be extended to include k = 9 but we show that

*in*every

*graph*with circumference nine we can destroy all 9-

*cycles*by

*removing*1/5

*of*the vertices. ... Acknowledgment All the authors wish to thank the University

*of*South Africa and the National Research Foundation

*of*South Africa for sponsoring a workshop at Salt Rock, South Africa (23 March -

*4*April ...

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