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A note on path-perfect graphs

John Frederick Fink, H.Joseph Straight
1981 Discrete Mathematics  
It is noted that the< graphs I&, K(r, 2r -1) and K(r, 2r+ ?I are path-perfect. Also some resuk are given concerning the existen* of regular path-perfect graphs.  ...  In this paper we explore the c:oncept of factoring a graph into non-isomorphic paths. Lel Pi denote the path of length i.  ...  A lstence of re; ;ular pathperfect graphs -wk en 1' is odd and r divides neither n l;lor n + 1. Rx example, doses there exist a 15regular path-perfect graph of 019er 25 and size 210'?  ... 
doi:10.1016/0012-365x(81)90262-4 fatcat:tqguvyeltve4nlblukdraa2hha

Average connectivity and average edge-connectivity in graphs

Jaehoon Kim, Suil O
2013 Discrete Mathematics  
In addition, we show that this family has the fewest perfect matchings among cubic graphs that have perfect matchings.  ...  Connectivity and edge-connectivity of a graph measure the difficulty of breaking the graph apart, but they are very much affected by local aspects like vertex degree.  ...  Recall that every vertex in R has exactly one neighbor in S. For the vertex after u on a u, v-path, at most one vertex of S is available.  ... 
doi:10.1016/j.disc.2013.05.024 fatcat:watglpdtynbprf3fdrnjqdvvgy

Partial and perfect path covers of cographs

D.G. Kirkpatrick, K.Madhukar Reddy, C.Pandu Rangan, A. Srinivasan
1998 Discrete Applied Mathematics  
A set r/P of disjoint paths in a graph G is called a (complete) path cover of G if every vertex of G belongs to one of the paths in :fl.  ...  In this paper, we introduce a variant of a minimum path cover, called a perfect path cover.  ...  Recall that our construction of a perfect path cover of a product graph depends only on the perfect path cover of its largest factor.  ... 
doi:10.1016/s0166-218x(98)00101-2 fatcat:jkjd65lctjdr7b6du7aumacyuy

On perfect colorings of infinite multipath graphs

M. A. Lisitsyna, S. V. Avgustinovich, O. G. Parshina
2020 Sibirskie Elektronnye Matematicheskie Izvestiya  
We consider a lexicographic product of the innite path graph and a graph G that can be either the complete or empty graph on n vertices.  ...  A coloring of vertices of a given graph is called perfect if the color structure of each sphere of radius 1 in the graph depends only on the color of the sphere center. Let n be a positive integer.  ...  There is a one-to-one correspondence between the perfect colorings of an innite path graph and the block-monochrome perfect colorings of C ∞ · K n and C ∞ · K n ; the latter are obtained by K n -times  ... 
doi:10.33048/semi.2020.17.139 fatcat:zou3m5yqrfguldtd7i2lsfe5ia

Extending Maximal Perfect Haplotype Blocks to the Realm of Pangenomics [chapter]

Lucia Williams, Brendan Mumey
2020 Lecture Notes in Computer Science  
We define a maximal perfect pangenome haplotype block and give a lineartime, suffix tree based approach to find all such blocks from a set of pangenome haplotypes.  ...  We demonstrate the method by applying it to a pangenome built from yeast strains.  ...  A bubble in a De Bruijn graph that represents a SNP; we arbitrarily consider one side of the bubble to be the '0' path and the other to be the '1' path.  ... 
doi:10.1007/978-3-030-42266-0_4 fatcat:on7iacahxnhivpauodh66cqdy4

Finding Induced Paths of Given Parity in Claw-Free Graphs [chapter]

Pim van 't Hof, Marcin Kamiński, Daniël Paulusma
2010 Lecture Notes in Computer Science  
Finally, we show that a shortest induced path of given parity between two specified vertices of a claw-free perfect graph can be found in O(n 7 ) time. An extended abstract of this paper has been  ...  The Parity Path problem is to decide if a given graph contains both an induced path of odd length and an induced path of even length between two specified vertices.  ...  They present an O(|G|m + n) time algorithm for the Group Path problem on chordal graphs using a perfect elimination ordering.  ... 
doi:10.1007/978-3-642-11409-0_30 fatcat:hstidpzhcjgvzka2vl6sd6yxma

Finding Induced Paths of Given Parity in Claw-Free Graphs

Pim van 't Hof, Marcin Kamiński, Daniël Paulusma
2010 Algorithmica  
Finally, we show that a shortest induced path of given parity between two specified vertices of a claw-free perfect graph can be found in O(n 7 ) time. An extended abstract of this paper has been  ...  The Parity Path problem is to decide if a given graph contains both an induced path of odd length and an induced path of even length between two specified vertices.  ...  They present an O(|G|m + n) time algorithm for the Group Path problem on chordal graphs using a perfect elimination ordering.  ... 
doi:10.1007/s00453-010-9470-5 fatcat:emwqsmiedbfrnc4qd35kei5xki

A note on internally disjoint alternating paths in bipartite graphs

Dingjun Lou, Akira Saito, Lihua Teng
2005 Discrete Mathematics  
Let G be a balanced bipartite graph with partite sets X and Y, which has a perfect matching, and let x ∈ X and y ∈ Y . Let k be a positive integer.  ...  Then we prove that if G has k internally disjoint alternating paths between x and y with respect to some perfect matching, then G has k internally disjoint alternating paths between x and y with respect  ...  For graph-theoretic terminology not defined in this note, we refer the reader to [2] . In this note, a path which starts from a vertex x and ends at a vertex y is called an xy-path.  ... 
doi:10.1016/j.disc.2004.10.019 fatcat:5rgbggxtr5cp7ffs5ekhaystli

Broder's Chain Is Not Rapidly Mixing [article]

Annabell Berger, Steffen Rechner
2014 arXiv   pre-print
We prove that Broder's Markov chain for approximate sampling near-perfect and perfect matchings is not rapidly mixing for Hamiltonian, regular, threshold and planar bipartite graphs, filling a gap in the  ...  graph is known.  ...  (Note that a ladder with 0 rungs has one perfect matching and a ladder with 1 rung, too.) We find |N uv (G)| = 6 3k F k+3 . Let us consider the set of perfect matchings M (G) in this graph.  ... 
arXiv:1404.4249v1 fatcat:t52tnnbmirha3oq3s2z2h6tgxe

Page 6607 of Mathematical Reviews Vol. , Issue 97K [page]

1997 Mathematical Reviews  
Haruo Hosoya (J-OCHS-I; Bunkyo) 97k:05164 05C70 Zhang, Fuji (PRC-XIAM; Xiamen); Zhang, Heping |Zhang, He Ping*| (PRC-LAN; Lanzhou) A note on the number of perfect matchings of bipartite graphs.  ...  of the paths modulo three, and then choosing (roughly) every third vertex on each path to form a dominating set.  ... 

Traveling salesman path problems

Fumei Lam, Alantha Newman
2006 Mathematical programming  
We first characterize traveling salesman walk perfect graphs, graphs for which the convex hull of incidence vectors of traveling salesman walks can be described by linear inequalities.  ...  The objective is to find a minimum cost path from the source to destination visiting all cities exactly once.  ...  Acknowledgements The authors would like to thank Santosh Vempala for suggesting problems on traveling salesman paths, Kevin Cheung, Michel Goemans, Kunal Talwar Santosh Vempala for many helpful discussions  ... 
doi:10.1007/s10107-006-0046-8 fatcat:6wa5ktitcrbi3mlt5l7vkwfdcu

On perfect colorings of infinite multipath graphs [article]

M. A. Lisitsyna, S. V. Avgustinovich, O. G. Parshina
2020 arXiv   pre-print
We consider a lexicographic product of the infinite path graph and a graph G that can be either the complete or empty graph on n vertices.  ...  A coloring of vertices of a given graph is called perfect if the color structure of each ball of radius 1 in the graph depends only on the color of the ball center. Let n be a positive integer.  ...  There is a one-to-one correspondence between the perfect colorings of an infinite path graph and the block monochrome perfect colorings of C ∞ · G; the latter are obtained by G-times copying of the former  ... 
arXiv:2003.06803v1 fatcat:kzgk5iq5lnef7a6oqfd3dunbku

The computational strength of matchings in countable graphs [article]

Stephen Flood, Matthew Jura, Oscar Levin, Tyler Markkanen
2020 arXiv   pre-print
In a 1977 paper, Steffens identified an elegant criterion for determining when a countable graph has a perfect matching.  ...  The results of this paper explore the relationship between graph theory and logic by showing the way in which specific changes to a single graph-theoretic principle impact the corresponding proof-theoretical  ...  To see why the new graph G , l , r , c has a unique perfect matching, note that in any perfect matching, exactly one of l or r matches into G • .  ... 
arXiv:2006.11334v1 fatcat:gnjsgfyngnajbkswztsoua2wdy

β-Perfect Graphs

S.E. Markossian, G.S. Gasparian, B.A. Reed
1996 Journal of combinatorial theory. Series B (Print)  
In particular, both referees noticed a large hole in an earlier erroneous proof of Theorem 4 and one asked about the complexity of determining if a graph is ;-perfect.  ...  Even Holes and ;-Perfect Graphs By a hole, we mean an induced subgraph isomorphic to a chordless cycle on at least four vertices. C k denotes a hole on k vertices.  ...  However, a ;-perfect graph can contain no chordless cycle on 2k vertices, k 2 (since ;(C 2k )=3 and /(C 2k )=2).  ... 
doi:10.1006/jctb.1996.0030 fatcat:bkhx6jcbaffyhpmkjeos2t6dai

Inverses of bipartite graphs

Yujun Yang, Dong Ye
2017 Combinatorica  
In this note, we characterize all bipartite graphs with a unique perfect matching whose adjacency matrices have inverses diagonally similar to non-negative matrices, which settles an open problem of Godsil  ...  on inverses of bipartite graphs in [Godsil, Inverses of Trees, Combinatorica 5 (1985) 33-39].  ...  : If P is an M -alternating path, then G\V (P ) has a perfect matching. So G\V (P ) has a Sachs subgraph. Now assume that G\V (P ) has a Sachs subgraph. Note that G\V (P ) is a bipartite graph.  ... 
doi:10.1007/s00493-016-3502-y fatcat:4onia7cvabcvjl44osk3gplud4
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