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

Rebecca Stones
2014 Discrete Mathematics & Theoretical Computer Science  
Analysis 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] .  ... 
doi:10.46298/dmtcs.1248 fatcat:pb2xwj4zhrh6zjsu7xikc3x4yi

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

Rebecca Stones
2014 Discrete Mathematics and Theoretical Computer Science DMTCS   unpublished
Let f m,n,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] .  ... 
fatcat:tog3oy437nbxbmeisf36foqh6q

Weighted Upper Edge Cover: Complexity and Approximability [chapter]

Kaveh Khoshkhah, Mehdi Khosravian Ghadikolaei, Jérôme Monnot, Florian Sikora
2018 Lecture Notes in Computer Science  
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 ∆ 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.  ... 
doi:10.1007/978-3-030-10564-8_19 fatcat:z6rt2hzvojfodc32dufdbji2q4

Weighted Upper Edge Cover: Complexity and Approximability [article]

Kaveh Khoshkhah, Mehdi Khosravian Ghadikolaei, Jerome Monnot and Florian Sikora
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.  ... 
arXiv:1811.02599v1 fatcat:zzobgty6wnawpn7x3hyxps2wnq

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

Edson Norberto Cacer, Henrique Mongelli, Christiane Nishibe, Siang Wun Song
2010 2010 International Conference on High Performance Computing & Simulation  
Dehne et al. present a BSP/CGM algorithm for 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  ... 
doi:10.1109/hpcs.2010.5547062 dblp:conf/ieeehpcs/CaceresMNS10 fatcat:pxxd3orv6fcehj4abjgwjozhp4

Dominating sets whose closed stars form spanning trees

Jerrold W. Grossman
1997 Discrete Mathematics  
For a subset W 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.  ... 
doi:10.1016/0012-365x(95)00334-s fatcat:i22kctg4jjdrtb5fjoebflizmm

Optimal per-edge processing times in the semi-streaming model

Mariano Zelke
2007 Information Processing Letters  
for any constant k and for 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.  ... 
doi:10.1016/j.ipl.2007.06.004 fatcat:auvnyeq7rzaifcvrrekr53qsfm

Optimal Per-Edge Processing Times in the Semi-Streaming Model [article]

Mariano Zelke
2007 arXiv   pre-print
respectively for any constant k and for 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.  ... 
arXiv:0708.4284v1 fatcat:dh7rlt3skbfbvncpffuoln2nsy

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]

Cristina Bazgan, Basile Couëtoux, Zsolt Tuza
2009 Lecture Notes in Computer Science  
Given an undirected 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.  ... 
doi:10.1007/978-3-642-10631-6_90 fatcat:pa52occsebcnnnplm734zmf3au

On the maximum number of copies of H in graphs with given size and order [article]

Dániel Gerbner, Dániel T. Nagy, Balázs Patkós, Máté Vizer
2018 arXiv   pre-print
We study 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.  ... 
arXiv:1810.00817v1 fatcat:ve42yvh2w5hpnbjktflnsxbotu

On spanning tree edge denpendences of graphs [article]

Yujun Yang, Can Xu
2022 arXiv   pre-print
Let τ(G) and τ_G(e) be 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  ... 
arXiv:2203.14566v1 fatcat:qlefhblsqvfttl3q5hrg5weplu

Research of NP-Complete Problems in the Class of Prefractal Graphs

Rasul Kochkarov
2021 Mathematics  
We propose a class 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.  ... 
doi:10.3390/math9212764 fatcat:kitwgsxezvdhfahfz5lhyrt2ca

Degree-preserving spanning trees in small-degree graphs

Peter Damaschke
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  ... 
doi:10.1016/s0012-365x(00)00005-4 fatcat:5e2mgucijzhnrha5hq6gbroi2q

A study on prefixes of c_2 invariants [article]

Karen Yeats
2018 arXiv   pre-print
These calculations support 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.  ... 
arXiv:1805.11735v2 fatcat:g5ivl73ecbcwrekaret2dvkiii
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