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### Removable Edges in Longest Cycles of 4-Connected Graphs

Jichang Wu, Xueliang Li
2004 Graphs and Combinatorics
Let G be a 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]

Asghar Asgharian Sardroud, Alireza Bagheri
2015 arXiv   pre-print
More precisely, our algorithm finds a 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

Hajo Broersma, Fedor V. Fomin, Pim van 't Hof, Daniël Paulusma
2011 Algorithmica
For a claw-free 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

Nili Guttmann-Beck, Refael Hassin
1998 Discrete Applied Mathematics
It runs 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,).  ...

### On the Intersections of Longest Cycles in a Graph

Iain A. Stewart, Ben Thompson
1995 Experimental Mathematics
The circumference 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]

Zh. G. Nikoghosyan
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 ≥ 4δ − 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]

Edita Máčajová, Ján Mazák
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

Igor Fabrici, Jochen Harant, Stanislav Jendrol'
2016 Discussiones Mathematicae Graph Theory
Moreover, new lower bounds on the length 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

D.A. Angel Sherin, V. Maheswari
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

Susan A. van Aardt, Alewyn P. Burger, Jean E. Dunbar, Marietjie Frick, Bernardo Llano, Carsten Thomassen, Rita Zuazua
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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