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A theorem on connected graphs in which every edge belongs to a 1-factor

Charles H. C. Little
1974 Journal of the Australian Mathematical Society  
A graph G is factor covered if and only if for every e, e' e E(G) there exists an alternating cycle containing both e and e'. PROOF.  ...  Let A^ be an alternating cycle containing e and e k and let A 2 be an alternating cycle containing e k and e'; A v and A 2 exist by the induction hypothesis.  ... 
doi:10.1017/s144678870002913x fatcat:dzabq5djabgkdel75fpr3novcu

Scaffolding Problems Revisited: Complexity, Approximation and Fixed Parameter Tractable Algorithms, and Some Special Cases

Mathias Weller, Annie Chateau, Clément Dallard, Rodolphe Giroudeau
2018 Algorithmica  
The problem consists in finding a collection of disjoint cycles and paths covering a particular graph called the "scaffold graph".  ...  We propose negative and positive results, exploring the frontier between difficulty and tractability of computing and/or approximating a solution to the problem.  ...  a collection S of ≤ σ p alternating paths and ≤ σ c alternating cycles and G can be covered by a collection S of ≤ σ p alternating paths and ≤ σ c alternating cycles such that ω(S) − ω(S ) ≥ l?  ... 
doi:10.1007/s00453-018-0405-x fatcat:m6eaidpx6remvjglwfxd7mnybm

Programs with minimal goto statements

Shigeki Iwata
1978 Information and Control  
We can apply the algorithm not only to an arbitrate loop free flowgraph but also to the class of "gate flowgraph" in which all entries and exits are particular for every cycle of the flowgraph.  ...  Yaku and Dr. T. Kasai for their helpful advice and to the referee who made many helpful comments. He would like to thank Dr. T.S. Han, Dr. T. Nishizeki and Miss H.  ...  Kadokura for his encouragement and to Professor K. Utsunomiya for his suggestion of this work. I~CEIVED: April 4, 1975; VJ~VlSED: June 17, 1977  ... 
doi:10.1016/s0019-9958(78)90473-4 fatcat:r4zi3mehyzahrnws5qzmm7pape

Approximation Algorithms for Two-Machine Flow-Shop Scheduling with a Conflict Graph [chapter]

Yinhui Cai, Guangting Chen, Yong Chen, Randy Goebel, Guohui Lin, Longcheng Liu, An Zhang
2018 Lecture Notes in Computer Science  
Path cover is a well-known intractable problem that finds a minimum number of vertex disjoint paths in a given graph to cover all the vertices.  ...  We further show that another variant, where the objective function is the total number of length-0 and length-1 paths, is also polynomial-time solvable.  ...  one edge per cycle, and return the resulting path cover.  ... 
doi:10.1007/978-3-319-94776-1_18 fatcat:l75nvwz3hbgkfj7dkvyu6lkwke

Parallel and sequential approximation of shortest superstrings [chapter]

Artur Czumaj, Leszek Gasieniec, Marek Piotrów, Wojciech Rytter
1994 Lecture Notes in Computer Science  
There are two basic measures of the approximations: the compression ratio and the approximation ratio. The well known and practical approximation algorithm is the sequential algorithm GREEDY.  ...  Previously it was reduced by Teng and Yao from 3 to 2.89. 4 The design of an RN C algorithm with constant approximation ratio and an N C algorithm with logarithmic approximation ratio.  ...  Let also a path-cycle cover be a collection of paths and cycles in G such that each vertex is in exactly one path or cycle.  ... 
doi:10.1007/3-540-58218-5_9 fatcat:5ejkgxfuhfhvfcp3mvtmrwyycm

Approximation algorithms for the directed path partition problems [article]

Yong Chen, Zhi-Zhong Chen, Curtis Kennedy, Guohui Lin, Yao Xu, An Zhang
2021 arXiv   pre-print
When k ≥ 7, we present an improved (k+2)/3-approximation algorithm based on the maximum path-cycle cover followed by a careful 2-cycle elimination process.  ...  Given a directed graph G = (V, E), the k-path partition problem is to find a minimum collection of vertex-disjoint directed paths each of order at most k to cover all the vertices of V.  ...  One sees that a 1-matching in the directed graph G = (V, E) consists of vertex-disjoint paths and cycles, and thus it is also called a path-cycle cover.  ... 
arXiv:2107.04699v1 fatcat:2uzy5rgxajfr3nxd6kjfh3y23y

Stabilizing Weighted Graphs [article]

Zhuan Khye Koh, Laura Sanità
2017 arXiv   pre-print
In particular, one of the main ingredients of our result is the development of a polynomial-time algorithm to compute a basic maximum-weight fractional matching with minimum number of odd cycles in its  ...  The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which are of independent interest.  ...  Since N (x) and N (x * ) are matchings in G, J is made up of vertex-disjoint paths and cycles of G. For each such path or cycle, its edges alternately belong to N (x) or N (x * ).  ... 
arXiv:1709.01982v2 fatcat:z6j62mronrdzjdk7el74raiike

Algorithmic proofs of two relations between connectivity and the 1-factors of a graph

Harold N. Gabow
1979 Discrete Mathematics  
Conditions for exactly n, or more than n, totally covered vertices are also given. Introdu~ou 5 *  ...  give simple proofs of these two known relations in the theory of matched graphs: A graph with a unique l-factor contains a matched bridge; an n-connected graph with a l-factor has at least n totally covered  ...  Throughout this paper, we consider only graphs that have a l-factor, An alternating path (cycle) is a simple path (cycle) whose edges are alternately matched and unmatched, If C is an alternating cycle  ... 
doi:10.1016/0012-365x(79)90058-x fatcat:j4vkvzrqc5hupl5ttmtvcve4qu

On the Existence of a Cyclic Near-Resolvable (6n+4)-Cycle System of 2K12n+9

Raja'i Aldiabat, Haslinda Ibrahim, Sharmila Karim
2019 Journal of Mathematics  
Then, some new properties and results related to this construction are formulated.  ...  In this article, we prove the existence of a simple cyclic near-resolvable v-1/2-cycle system of 2Kv for v≡9 mod 12 by the method of constructing its starter.  ...  Acknowledgments The authors are very grateful to the anonymous referees for their valuable comments and suggestions.  ... 
doi:10.1155/2019/5276753 fatcat:kaz6arx2cjgrdg4puxvgccsdmm

On strongly connected digraphs with bounded cycle length

Samir Khuller, Balaji Raghavachari, Neal Young
1996 Discrete Applied Mathematics  
This paper gives a proof that, for graphs where each directed cycle has at most three edges, the MEG problem is equivalent to maximum bipartite matching, and therefore solvable in polynomial time.  ...  Ravi and Klaus Truemper for referees for useful comments on a preliminary  ...  If it exists, the edge (u, u) is necessary, because any alternate path from u to v would have to have length two and all such paths have been checked. Otherwise, check all return paths of length two.  ... 
doi:10.1016/0166-218x(95)00105-z fatcat:joagp2v22zedjf3bq2utlfthda

Almost All Regular Graphs are Normal [article]

Seyed Saeed Changiz Rezaei, Seyyed Aliasghar Hosseini, Bojan Mohar
2015 arXiv   pre-print
In 1999, De Simone and Körner conjectured that every graph without induced C_5,C_7,C_7 contains a clique cover C and a stable set cover I such that every clique in C and every stable set in I have a vertex  ...  This conjecture has roots in information theory and became known as the Normal Graph Conjecture.  ...  paths and cycles.  ... 
arXiv:1511.07591v1 fatcat:ly44v3o7fvevtelicmtyqzmpcy

A 7/9 - Approximation Algorithm for the Maximum Traveling Salesman Problem [chapter]

Katarzyna Paluch, Marcin Mucha, Aleksander Ma̧dry
2009 Lecture Notes in Computer Science  
A path P is alternating with respect to a cycle cover C 1 if its edges are alternatingly from C 1 and from E \C 1 .  ...  If an alternating path ends and begins with the same vertex, then it is called an alternating cycle.  ...  Appendix 5 Algorithm for well 2-almost-cycle-coloring is equal to exactly the alternating weight of the fragment connected with v ′ 1 , v ′ 2 .  ... 
doi:10.1007/978-3-642-03685-9_23 fatcat:femdrvuzevayjlw6eyv72hktzi

Stabilizing Weighted Graphs

Zhuan Khye Koh, Laura Sanità, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming  
In particular, one of the main ingredients of our result is the development of a polynomial-time algorithm to compute a basic maximum-weight fractional matching with minimum number of odd cycles in its  ...  The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which are of independent interest.  ...  Since N (x) and N (x * ) are matchings in G, J is made up of vertex-disjoint paths and cycles of G. For each such path or cycle, its edges alternately belong to N (x) or N (x * ).  ... 
doi:10.4230/lipics.icalp.2018.83 dblp:conf/icalp/KohS18 fatcat:yg2xxp5uorbajlj2grkt3doijq

Alternating Paths and Cycles of Minimum Length [chapter]

William S. Evans, Giuseppe Liotta, Henk Meijer, Stephen Wismath
2015 Lecture Notes in Computer Science  
We extend our approach to compute shortest alternating paths in O(n 2 ) time with two bends per edge and to compute shortest alternating cycles on 3-colored pointsets in O(n 2 ) time with O(n) bends per  ...  We study the problem of computing a planar drawing of a cycle of minimum length that contains vertices at points R ∪ B and alternates colors.  ...  Research also supported by NSERC, and by MIUR of Italy under project AlgoDEEP prot. 2008TFBWL4.  ... 
doi:10.1007/978-3-319-27261-0_32 fatcat:xolzh74urrc6xhtz76fbiq5qfm

Alternating cycles and trails in 2-edge-coloured complete multigraphs

Jørgen Bang-Jensen, Gregory Gutin
1998 Discrete Mathematics  
Bankfalvi and Bankfalvi (1968) obtained a necessary and sufficient condition for G to have a Hamiltonian alternating cycle.  ...  Our result is a generalization of a theorem by Saad (1996) that determines the length of a longest alternating cycle in G.  ...  Tort and the referees for their helpful remarks and suggestions.  ... 
doi:10.1016/s0012-365x(97)00274-4 fatcat:bozxpfcgnnccrhc4loojwyzpii
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