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Computing the number of h-edge spanning forests in complete bipartite graphs

2014
*
Discrete Mathematics & Theoretical Computer Science
*

Analysis

doi:10.46298/dmtcs.1248
fatcat:pb2xwj4zhrh6zjsu7xikc3x4yi
*of*Algorithms International audience Let fm,n,*h*be*the**number**of**spanning**forests*with*h**edges**in**the**complete**bipartite**graph*Km,n. ... Kirchhoff\textquoterights Matrix Tree Theorem implies fm,n,m+n-1=mn-1 nm-1 when m ≥1 and n ≥1, since fm,n,m+n-1 is*the**number**of**spanning*trees*in*Km,n. ... Acknowledgements*The*author would like to thank Graham Farr for his feedback.*The*author would also like to thank Zur Luria for directing her to [3] . ...##
###
Computing the number of h-edge spanning forests in complete bipartite graphs

2014
*
Discrete Mathematics and Theoretical Computer Science DMTCS
*
unpublished

Let f m,n,

fatcat:tog3oy437nbxbmeisf36foqh6q
*h*be*the**number**of**spanning**forests*with*h**edges**in**the**complete**bipartite**graph*Km,n. ... Kirchhoff's Matrix Tree Theorem implies fm,n,m+n−1 = m n−1 n m−1 when m ≥ 1 and n ≥ 1, since fm,n,m+n−1 is*the**number**of**spanning*trees*in*Km,n. ... Acknowledgements*The*author would like to thank Graham Farr for his feedback.*The*author would also like to thank Zur Luria for directing her to [3] . ...##
###
Weighted Upper Edge Cover: Complexity and Approximability
[chapter]

2018
*
Lecture Notes in Computer Science
*

We show that

doi:10.1007/978-3-030-10564-8_19
fatcat:z6rt2hzvojfodc32dufdbji2q4
*the*weighted Upper*Edge*Cover is much more difficult than Upper*Edge*Cover because it is not O( 1 n 1/2−ε ) approximable, nor O( 1 ∆ 1−ε )*in**edge*-weighted*graphs**of*size n and maximmm degree ...*In*this paper, we propose to study*the*weighted version*of*Maximum Minimal*Edge*Cover called Upper*Edge*Cover, a problem having application*in*genomic sequence alignment. ...*The*degree d G (v)*of*vertex v ∈ V*in*G is*the**number**of**edges*incident to v and ∆(G) is*the*maximum degree*of**the**graph*G. ...##
###
Weighted Upper Edge Cover: Complexity and Approximability
[article]

2018
*
arXiv
*
pre-print

*In*this paper, we propose to study

*the*weighted version

*of*Maximum Minimal

*Edge*Cover called Upper

*Edge*Cover, a problem having application

*in*

*the*genomic sequence alignment. ... We show that

*the*weighted Upper

*Edge*Cover is much more difficult than Upper

*Edge*Cover because it is not O(1/n^1/2-ε) approximable, nor O(1/Δ^1-ε)

*in*

*edge*-weighted

*graphs*

*of*size n and maximum degree ...

*The*degree d G (v)

*of*vertex v ∈ V

*in*G is

*the*

*number*

*of*

*edges*incident to v and ∆(G) is

*the*maximum degree

*of*

*the*

*graph*G. ...

##
###
Experimental results of a coarse-grained parallel algorithm for spanning tree and connected components

2010
*
2010 International Conference on High Performance Computing & Simulation
*

Dehne et al. present a BSP/CGM algorithm for

doi:10.1109/hpcs.2010.5547062
dblp:conf/ieeehpcs/CaceresMNS10
fatcat:pxxd3orv6fcehj4abjgwjozhp4
*computing*a*spanning*tree and*the*connected components*of*a*graph*, that requires O(log p) communication rounds, where p is*the**number**of*processors. ... It requires*the*solution*of**the*Euler tour problem which*in*turn is based on*the*solution*of**the*list ranking problem. ... ACKNOWLEDGMENTS We thank*the*referees for their comments. This work has been partially supported by CNPq Proc. No ...##
###
Dominating sets whose closed stars form spanning trees

1997
*
Discrete Mathematics
*

For a subset W

doi:10.1016/0012-365x(95)00334-s
fatcat:i22kctg4jjdrtb5fjoebflizmm
*of*vertices*of*an undirected*graph*G, let S(W) be*the*subgraph consisting*of*W, all*edges*incident to at least one vertex*in*W, and all vertices adjacent to at least one vertex*in*W. ... Among*the*results*in*this paper are a characterization*of*those values*of*n and m for which there exists a connected*graph*with n vertices and m*edges*that has no*spanning*star tree, and a proof that finding ... Acknowledgements*The*author thanks Suzanne Zeitman for helpful conversations on several aspects*of*this paper. ...##
###
Optimal per-edge processing times in the semi-streaming model

2007
*
Information Processing Letters
*

for any constant k and for

doi:10.1016/j.ipl.2007.06.004
fatcat:auvnyeq7rzaifcvrrekr53qsfm
*computing*a minimum*spanning**forest*O(log n). ... Every presented algorithm determines a solution asymptotically as fast as*the*best corresponding algorithm up to date*in**the*classical RAM model, which therefore cannot convert*the*advantage*of*unlimited ...*In*[4] semi-streaming algorithms are given for*computing**the*connected components and a*bipartition**of*a*graph*as well as a minimum*spanning*tree*of*a weighted*graph*. ...##
###
Optimal Per-Edge Processing Times in the Semi-Streaming Model
[article]

2007
*
arXiv
*
pre-print

respectively for any constant k and for

arXiv:0708.4284v1
fatcat:dh7rlt3skbfbvncpffuoln2nsy
*computing*a minimum*spanning**forest*O(log n). ... Every presented algorithm determines a solution asymptotically as fast as*the*best corresponding algorithm up to date*in**the*classical RAM model, which therefore cannot convert*the*advantage*of*unlimited ...*In*[4] semi-streaming algorithms are given for*computing**the*connected components and a*bipartition**of*a*graph*as well as a minimum*spanning*tree*of*a weighted*graph*. ...##
###
Page 5051 of Mathematical Reviews Vol. , Issue 82m
[page]

1982
*
Mathematical Reviews
*

*The*author considers decomposition and balanced decomposition

*of*

*the*

*complete*multipartite

*graph*into copies

*of*

*complete*

*bipartite*

*graphs*;

*the*latter requires each vertex to occur

*in*

*the*same

*number*

*of*... If G is a mixed

*graph*(with both directed and nondirected

*edges*), a matching

*forest*

*in*G is a

*forest*J such that, for every vertex v,

*the*

*number*

*of*directed

*edges*

*of*J directed towards v plus

*the*82m:05075b ...

##
###
Covering a Graph with a Constrained Forest (Extended Abstract)
[chapter]

2009
*
Lecture Notes in Computer Science
*

Given an undirected

doi:10.1007/978-3-642-10631-6_90
fatcat:pa52occsebcnnnplm734zmf3au
*graph*on n vertices with weights on its*edges*, Min WCF(p) consists*of**computing*a covering*forest**of*minimum weight such that each*of*its tree components contains at least p vertices ... We prove here that for any p ≥ 4,*the*unweighted version is NP -hard, even for planar*bipartite**graphs**of*maximum degree 3; moreover,*the*unweighted version for any p ≥ 3 has no ptas for*bipartite**graphs*... We can do this by artificially*completing*F with sets E blue , E red*of*blue and red*edges*(not necessarily*edges**of*G) to obtain a*forest*, each tree component*of*which*spans*a class*of*P. ...##
###
On the maximum number of copies of H in graphs with given size and order
[article]

2018
*
arXiv
*
pre-print

We study

arXiv:1810.00817v1
fatcat:ve42yvh2w5hpnbjktflnsxbotu
*the*maximum*number*ex(n,e,*H*)*of*copies*of*a*graph**H**in**graphs*with given*number**of*vertices and*edges*. ... We also investigate a variant*of*this problem, when*the*host*graph*is*bipartite*. ... that*the**number**of*these*spanning**forests*is asymptotically maximized by*the**graphs*stated*in**the*theorems. ...##
###
On spanning tree edge denpendences of graphs
[article]

2022
*
arXiv
*
pre-print

Let τ(G) and τ_G(e) be

arXiv:2203.14566v1
fatcat:qlefhblsqvfttl3q5hrg5weplu
*the**number**of**spanning*trees*of*a connected*graph*G and*the**number**of**spanning*trees*of*G containing*edge*e. ...*The*ratio d_G(e)=τ_G(e)/τ(G) is called*the**spanning*tree*edge*density*of*e, or simply density*of*e. ... For example,*the**number**of**spanning*trees have been*computed*for*complete**bipartite**graphs*[4, 5] ,*complete*multipartite*graphs*[6, 7] , cubic cycle C 3 N and*the*quadruple cycle C 4 N [9] ,*graphs*...##
###
Research of NP-Complete Problems in the Class of Prefractal Graphs

2021
*
Mathematics
*

We propose a class

doi:10.3390/math9212764
fatcat:kitwgsxezvdhfahfz5lhyrt2ca
*of*prefractal*graphs*and review particular statements*of*NP-*complete*problems. As an example, algorithms for searching for*spanning*trees and packing*bipartite**graphs*are proposed. ... NP-*complete*problems*in**graphs*, such as enumeration and*the*selection*of*subgraphs with given characteristics, become especially relevant for large*graphs*and networks. ... Conflicts*of*Interest:*The*author declares no conflict*of*interest. ...##
###
Degree-preserving spanning trees in small-degree graphs

2000
*
Discrete Mathematics
*

*In*

*the*degree-preserving

*spanning*tree problem, we seek a

*spanning*tree with a maximum

*number*

*of*vertices whose incident

*edges*all belong to

*the*

*spanning*tree. ... We prove NP-

*completeness*for

*bipartite*planar degree-5

*graphs*and for planar degree-3

*graphs*. This strengthens a result from

*the*mentioned paper. ...

*The*degree-preserving

*spanning*tree problem is to ÿnd,

*in*a given connected

*graph*G = (V; E), a

*spanning*tree T = (V; F) with

*the*maximum

*number*

*of*degree-preserving vertices, that means, all

*edges*

*of*...

##
###
A study on prefixes of c_2 invariants
[article]

2018
*
arXiv
*
pre-print

These calculations support

arXiv:1805.11735v2
fatcat:g5ivl73ecbcwrekaret2dvkiii
*the*idea that all possible finite sequences appear as initial segments*of*c_2 invariants,*in*contrast to their apparent sparsity on small*graphs*. ... Then it proceeds to report on some recent calculations*of*c_2 invariants for two families*of*circulant*graphs*at small primes. ... Given a*spanning**forest*polynomial on a*graph**H*and a set S ⊆ E(*H*), any assignment*of**the**edges**of*S yields a sum*of**spanning**forest*polynomials on*the**graph**H*− S. ...
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