Shorter Implicit Representation for Planar Graphs and Bounded Treewidth Graphs

2007 ◽  
pp. 582-593 ◽  
Author(s):  
Cyril Gavoille ◽  
Arnaud Labourel
Keyword(s):  
Planar Graphs ◽  
Implicit Representation ◽  
Bounded Treewidth

scholarly journals THE POINT-SET EMBEDDABILITY PROBLEM FOR PLANE GRAPHS

2013 ◽  
Vol 23 (04n05) ◽  
pp. 357-395 ◽  
Author(s):  
THERESE BIEDL ◽  
MARTIN VATSHELLE
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Line Drawing ◽  
Np Hard ◽  
Bounded Treewidth ◽  
Straight Line ◽  
Triangulated Graphs ◽  
Point Set ◽  
Fixed Embedding ◽  
Set Of Points

In this paper, we study the point-set embeddability problem, i.e., given a planar graph and a set of points, is there a mapping of the vertices to the points such that the resulting straight-line drawing is planar? It was known that this problem is NP-hard if the embedding can be chosen, but becomes polynomial for triangulated graphs of treewidth 3. We show here that in fact it can be answered for all planar graphs with a fixed combinatorial embedding that have constant treewidth and constant face-degree. We prove that as soon as one of the conditions is dropped (i.e., either the treewidth is unbounded or some faces have large degrees), point-set embeddability with a fixed embedding becomes NP-hard. The NP-hardness holds even for a 3-connected planar graph with constant treewidth, triangulated planar graphs, or 2-connected outer-planar graphs. These results also show that the convex point-set embeddability problem (where faces must be convex) is NP-hard, but we prove that it becomes polynomial if the graph has bounded treewidth and bounded maximum degree.

A Note on the Minimum H-Subgraph Edge Deletion

2015 ◽  
Vol 26 (03) ◽  
pp. 399-411
Author(s):  
Alexander Grigoriev ◽  
Bert Marchal ◽  
Natalya Usotskaya
Keyword(s):  
Planar Graphs ◽  
General Framework ◽  
Approximation Scheme ◽  
Linear Time ◽  
Time Algorithm ◽  
Polynomial Time Approximation Scheme ◽  
Edge Deletion ◽  
Time Dynamic ◽  
Bounded Treewidth ◽  
Programming Algorithms

In this note we consider the following problem: Given a graph [Formula: see text] and a subgraph [Formula: see text], what is the smallest subset [Formula: see text] of edges in [Formula: see text] that needs to be deleted from the graph to make it [Formula: see text]-free? Several algorithmic results are presented. First, using the general framework of Courcelle [9], we show that, for a fixed subgraph [Formula: see text], the problem can be solved in linear time on graphs of bounded treewidth. It is known that the constant hidden in the big-O notation of Courcelle algorithm is big which makes the approach impractical. Thus, we present two explicit linear time dynamic programming algorithms on graphs of bounded treewidth for restricted settings of the problem with reasonable constants. Third, using the linear time algorithm for graphs of bounded treewidth, we design a Baker's type polynomial time approximation scheme for the problem on planar graphs.

scholarly journals Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth

2011 ◽  
Vol 58 (5) ◽  
pp. 1-37 ◽  
Author(s):  
Mohammadhossein Bateni ◽  
Mohammadtaghi Hajiaghayi ◽  
Dániel Marx
Keyword(s):  
Planar Graphs ◽  
Approximation Schemes ◽  
Bounded Treewidth ◽  
Steiner Forest

Even-hole-free planar graphs have bounded treewidth

2008 ◽  
Vol 30 ◽  
pp. 129-134
Author(s):  
Aline Alves da Silva ◽  
Ana Silva ◽  
Cláudia Linhares Sales
Keyword(s):  
Planar Graphs ◽  
Bounded Treewidth

Acute Constraints in Straight-Line Drawings of Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 994-1001
Author(s):  
Akane SETO ◽  
Aleksandar SHURBEVSKI ◽  
Hiroshi NAGAMOCHI ◽  
Peter EADES
Keyword(s):  
Planar Graphs ◽  
Line Drawings ◽  
Straight Line

A Space-Efficient Separator Algorithm for Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 1007-1016 ◽  
Author(s):  
Ryo ASHIDA ◽  
Sebastian KUHNERT ◽  
Osamu WATANABE
Keyword(s):  
Planar Graphs

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

scholarly journals Maximal distance spectral radius of 4-chromatic planar graphs

2021 ◽  
Vol 618 ◽  
pp. 183-202
Author(s):  
Aysel Erey
Keyword(s):  
Spectral Radius ◽  
Planar Graphs ◽  
Maximal Distance

Note on (semi-)proper orientation of some triangulated planar graphs

2021 ◽  
Vol 392 ◽  
pp. 125723
Author(s):  
Ruijuan Gu ◽  
Hui Lei ◽  
Yulai Ma ◽  
Zhenyu Taoqiu
Keyword(s):  
Planar Graphs ◽  
Proper Orientation
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