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Approximation Algorithms for the Maximum Induced Planar and Outerplanar Subgraph Problems

Kerri Morgan, Graham Farr
2007 Journal of Graph Algorithms and Applications  
Efficient algorithms for finding large induced outerplanar graphs are also given.  ...  The results presented in this paper indicate that most existing algorithms perform substantially better than the existing lower bounds indicate.  ...  and palm trees.  ... 
doi:10.7155/jgaa.00141 fatcat:472ryjbh25amvkdipm5uourhum

k-Outerplanar Graphs, Planar Duality, and Low Stretch Spanning Trees [chapter]

Yuval Emek
2009 Lecture Notes in Computer Science  
Among the classes of graphs that received special attention in this context are k-outerplanar graphs (for a fixed k): Chekuri, Gupta, Newman, Rabinovich, and Sinclair show that every k-outerplanar graph  ...  In this paper it is shown that every k-outerplanar multigraph admits a spanning tree with constant average stretch.  ...  Acknowledgments I would like to thank Robert Krauthgamer, Manor Mendel, and David Peleg for helpful discussions.  ... 
doi:10.1007/978-3-642-04128-0_18 fatcat:4rxyuc65urgf5n3e6nwacfmdfm

k-Outerplanar Graphs, Planar Duality, and Low Stretch Spanning Trees

Yuval Emek
2010 Algorithmica  
Among the classes of graphs that received special attention in this context are k-outerplanar graphs (for a fixed k): Chekuri, Gupta, Newman, Rabinovich, and Sinclair show that every k-outerplanar graph  ...  In this paper it is shown that every k-outerplanar multigraph admits a spanning tree with constant average stretch.  ...  Acknowledgments I would like to thank Robert Krauthgamer, Manor Mendel, and David Peleg for helpful discussions.  ... 
doi:10.1007/s00453-010-9423-z fatcat:p4fmmayrgrczpcdllv3vs4bm3q

Induced Subgraphs of Bounded Degree and Bounded Treewidth [chapter]

Prosenjit Bose, Vida Dujmović, David R. Wood
2005 Lecture Notes in Computer Science  
In both these cases, our bounds on the order of H are tight. For bounded degree independent sets in trees, we characterise the extremal graphs.  ...  We prove that for all 0 ≤ t ≤ k and d ≥ 2k, every graph G with treewidth at most k has a 'large' induced subgraph H, where H has treewidth at most t and every vertex in H has degree at most d in G.  ...  That is, every n-vertex outerplanar graph G has an degree-6 independent set on 1 4 n − O(1) vertices, and for infinitely many values of n, there is an n-vertex outerplanar graph in which at most 1 4 n  ... 
doi:10.1007/11604686_16 fatcat:otychvzqsnftbp7n2aanmd7j2a

Vertex colorings without rainbow subgraphs

Wayne Goddard, Honghai Xu
2016 Discussiones Mathematicae Graph Theory  
For example, we show that the K 3 -upper chromatic number of any maximal outerplanar graph on n vertices is n/2 + 1.  ...  Given a coloring of the vertices of a graph G, we say a subgraph is rainbow if its vertices receive distinct colors.  ...  For m = 2, the rooks graph has no triangle, whence the result. In general, m + 1 is a lower bound by the independence number bound.  ... 
doi:10.7151/dmgt.1896 fatcat:hpkujxnpnfghfdb6sk34x4tfky

Induced Subgraphs of Bounded Degree and Bounded Treewidth [article]

Prosenjit Bose, Vida Dujmovic, David R. Wood
2005 arXiv   pre-print
In both these cases, our bounds on the order of H are tight. For bounded degree independent sets in trees, we characterise the extremal graphs.  ...  We prove that for all 0≤ t≤ k and d≥ 2k, every graph G with treewidth at most k has a 'large' induced subgraph H, where H has treewidth at most t and every vertex in H has degree at most d in G.  ...  Outerplanar Graphs A plane embedding of a graph in which every vertex is on a single face is called outerplanar. A graph is outerplanar if it has an outerplanar embedding.  ... 
arXiv:math/0505415v1 fatcat:kbz7wno4trdzbpltma7igvu7dy

Regular independent sets [article]

Yair Caro, Adriana Hansberg, Ryan Pepper
2015 arXiv   pre-print
In this article, we generalize and extend these results to find lower bounds for the regular k-independence number for trees, forests, planar graphs, k-trees and k-degenerate graphs.  ...  Lower bounds were proven for this invariant, in terms of the order, for trees and planar graphs.  ...  In a similar way as for the k-trees, we can compute lower bounds on α reg (G) for planar graphs, outerplanar graphs and k-degenerate graphs.  ... 
arXiv:1306.5026v3 fatcat:z6s5ysxabjaqhfps4qpf4izira

On the induced matching problem

Iyad Kanj, Michael J. Pelsmajer, Marcus Schaefer, Ge Xia
2011 Journal of computer and system sciences (Print)  
We derive similar results for outerplanar graphs and graphs of bounded genus. These extremal results can be applied to the area of parameterized computation.  ...  For example, we show that the induced matching problem on planar graphs has a kernel of size at most 40k that is computable in linear time; this significantly improves the results of Moser and Sikdar (  ...  Acknowledgments: We would like to thank the referee for valuable comments and suggestions that helped improve the quality and the presentation of the paper.  ... 
doi:10.1016/j.jcss.2010.09.001 fatcat:c6dfh3yigvd4dfvvxcugu5m53i

Repetition Number of Graphs

Yair Caro, Douglas B. West
2009 Electronic Journal of Combinatorics  
For line graphs of maximal outerplanar graphs, trees with perfect matchings, or triangulations with 2-factors, the lower bound is linear.  ...  Equality can hold when $2d$ is an integer, and the bound is approximately sharp in general, even when $G$ is restricted to be a tree, maximal outerplanar graph, planar triangulation, or claw-free graph  ...  Line graphs of sparse graphs For various families of sparse graphs with m edges, such as trees, maximal outerplanar graphs, and triangulations, we can improve the lower bounds on the repetition number  ... 
doi:10.37236/96 fatcat:dwqifqpasbg6tb7vd3rsu27v3q

Designing Low-Congestion Networks with Structural Graph Theory

Yota OTACHI
2011 Interdisciplinary Information Sciences  
We apply some newest results in Graph Minor Theory to the problem, and obtain efficient (even best possible) algorithms.  ...  In these lecture notes, we study problems of designing low-congestion subnetworks of given networks.  ...  Let T be a spanning tree of G that minimizes the congestion. We shall consider the average congestion of the edges of T, which is a lower bound of the congestion of T.  ... 
doi:10.4036/iis.2011.197 fatcat:tzxj4vzsbvacndnb4mofpvrjz4

Experimental evaluation of a tree decomposition-based algorithm for vertex cover on planar graphs

Jochen Alber, Frederic Dorn, Rolf Niedermeier
2005 Discrete Applied Mathematics  
on planar graphs, among others VERTEX COVER, in time O(c √ k n).  ...  In this paper, we present an experimental study for such tree decomposition-based algorithms focusing on VERTEX COVER.  ...  de Recerca Matemàtica, Barcelona, November [15] [16] [17] 2001, where, in particular, Robin Thomas and Ioan Todinca gave valuable comments and hints.  ... 
doi:10.1016/j.dam.2004.01.013 fatcat:oc2qwywki5brxgfbxuutplbzdi

Page 5581 of Mathematical Reviews Vol. , Issue 96i [page]

1996 Mathematical Reviews  
For these and certain other problems, the proof of solvability on k-outerplanar graphs also enlarges the class of planar graphs for which the problems are known to be solvable in polynomial time.” 96i:  ...  For fixed k, the problems of interest are solvable optimally in linear time on k-outerplanar graphs by dynamic programming.  ... 

I/O-Optimal Algorithms for Outerplanar Graphs

Anil Maheshwari, Norbert Zeh
2004 Journal of Graph Algorithms and Applications  
We also show that it takes O(sort(N )) I/Os to test whether a given graph is outerplanar and to compute an outerplanar embedding of an outerplanar graph, thereby providing O(sort(N ))-I/O algorithms for  ...  Our algorithms for the above problems use a simple linear-I/O time-forward processing algorithm for rooted trees whose vertices are stored in preorder.  ...  tree of an outerplanar graph G, independent of which representation of the output is chosen, we are only able to prove such a lower bound, if we place certain restrictions on the output representation  ... 
doi:10.7155/jgaa.00082 fatcat:rmcyrz6qyfgabpbc7l4lsanjga

I/O-Optimal Algorithms for Outerplanar Graphs [chapter]

Anil Maheshwari, Norbert Zeh
2006 Graph Algorithms and Applications 5  
We also show that it takes O(sort(N )) I/Os to test whether a given graph is outerplanar and to compute an outerplanar embedding of an outerplanar graph, thereby providing O(sort(N ))-I/O algorithms for  ...  Our algorithms for the above problems use a simple linear-I/O time-forward processing algorithm for rooted trees whose vertices are stored in preorder.  ...  tree of an outerplanar graph G, independent of which representation of the output is chosen, we are only able to prove such a lower bound, if we place certain restrictions on the output representation  ... 
doi:10.1142/9789812773289_0005 fatcat:umf5ff7saba3dejo7sf4qcrvoq

Fixed Parameter Algorithms for Planar Dominating Set and Related Problems [chapter]

Jochen Alber, Hans L. Bodlaender, Henning Fernau, Rolf Niedermeier
2000 Lecture Notes in Computer Science  
To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O( γ(G)), and that such a tree decomposition can be found in O( γ(G)n) time.  ...  Similar results can be obtained in the planar case for some variants of k-dominating set, e.g., k-independent dominating set and k-weighted dominating set.  ...  We are indebted to Ton Kloks for starting this research with us and for many discussions on the topic of this paper.  ... 
doi:10.1007/3-540-44985-x_10 fatcat:mibcmd5j75cwzoyqxuovbj2q6e
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