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

1974
*
Journal of the Australian Mathematical Society
*

A graph G is factor

doi:10.1017/s144678870002913x
fatcat:dzabq5djabgkdel75fpr3novcu
*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. ...##
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Scaffolding Problems Revisited: Complexity, Approximation and Fixed Parameter Tractable Algorithms, and Some Special Cases

2018
*
Algorithmica
*

The problem consists in finding a collection of disjoint

doi:10.1007/s00453-018-0405-x
fatcat:m6eaidpx6remvjglwfxd7mnybm
*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? ...##
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Programs with minimal goto statements

1978
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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

doi:10.1016/s0019-9958(78)90473-4
fatcat:r4zi3mehyzahrnws5qzmm7pape
*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 ...##
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Approximation Algorithms for Two-Machine Flow-Shop Scheduling with a Conflict Graph
[chapter]

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*. ...

##
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Parallel and sequential approximation of shortest superstrings
[chapter]

1994
*
Lecture Notes in Computer Science
*

There are two basic measures of the approximations: the compression ratio

doi:10.1007/3-540-58218-5_9
fatcat:5ejkgxfuhfhvfcp3mvtmrwyycm
*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*. ...##
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Approximation algorithms for the directed path partition problems
[article]

2021
*
arXiv
*
pre-print

When k ≥ 7, we present an improved (k+2)/3-approximation algorithm based on the maximum

arXiv:2107.04699v1
fatcat:2uzy5rgxajfr3nxd6kjfh3y23y
*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*. ...##
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Stabilizing Weighted Graphs
[article]

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

arXiv:1709.01982v2
fatcat:z6j62mronrdzjdk7el74raiike
*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 * ). ...##
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Algorithmic proofs of two relations between connectivity and the 1-factors of a graph

1979
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Discrete Mathematics
*

Conditions for exactly n, or more than n, totally

doi:10.1016/0012-365x(79)90058-x
fatcat:j4vkvzrqc5hupl5ttmtvcve4qu
*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*...##
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On the Existence of a Cyclic Near-Resolvable (6n+4)-Cycle System of 2K12n+9

2019
*
Journal of Mathematics
*

Then, some new properties

doi:10.1155/2019/5276753
fatcat:kaz6arx2cjgrdg4puxvgccsdmm
*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. ...##
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On strongly connected digraphs with bounded cycle length

1996
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Discrete Applied Mathematics
*

This paper gives a proof that, for graphs where each directed

doi:10.1016/0166-218x(95)00105-z
fatcat:joagp2v22zedjf3bq2utlfthda
*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. ...##
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Almost All Regular Graphs are Normal
[article]

2015
*
arXiv
*
pre-print

In 1999, De Simone

arXiv:1511.07591v1
fatcat:ly44v3o7fvevtelicmtyqzmpcy
*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*. ...##
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A 7/9 - Approximation Algorithm for the Maximum Traveling Salesman Problem
[chapter]

2009
*
Lecture Notes in Computer Science
*

A

doi:10.1007/978-3-642-03685-9_23
fatcat:femdrvuzevayjlw6eyv72hktzi
*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 . ...##
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Stabilizing Weighted Graphs

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

doi:10.4230/lipics.icalp.2018.83
dblp:conf/icalp/KohS18
fatcat:yg2xxp5uorbajlj2grkt3doijq
*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 * ). ...##
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Alternating Paths and Cycles of Minimum Length
[chapter]

2015
*
Lecture Notes in Computer Science
*

We extend our approach to compute shortest

doi:10.1007/978-3-319-27261-0_32
fatcat:xolzh74urrc6xhtz76fbiq5qfm
*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. ...##
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Alternating cycles and trails in 2-edge-coloured complete multigraphs

1998
*
Discrete Mathematics
*

Bankfalvi

doi:10.1016/s0012-365x(97)00274-4
fatcat:bozxpfcgnnccrhc4loojwyzpii
*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. ...
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