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### Tight Bounds for Linkages in Planar Graphs [chapter]

Isolde Adler, Stavros G. Kolliopoulos, Philipp Klaus Krause, Daniel Lokshtanov, Saket Saurabh, Dimitrios Thilikos
2011 Lecture Notes in Computer Science
In this paper we prove this result for planar graphs achieving g(k) = 2 O(k) . Our bound is radically better than the bounds known for general graphs.  ...  This fact is based on the celebrated Unique Linkage Theorem, whose -very technical -proof gives a function g(k) that is responsible for an immense parameter dependence in the running time of the algorithm  ...  We thank Ken-ichi Kawarabayashi and Paul Wollan for providing details on the bounds in  .  ...

### A lower bound for the tree-width of planar graphs with vital linkages [article]

2010 arXiv   pre-print
We give single exponential lower bounds both for the tree-width of planar graphs with vital linkages, and for the size of the grid necessary for finding irrelevant vertices.  ...  The algorithm uses a bound on the tree-width of graphs with vital linkages, and deletion of irrelevant vertices.  ...  In this paper we give a lower bound by showing that a (2 k + 1) × (2 k + 1) grid may not suffice -even in planar graphs. Despite recent progress  , the quest for good upper bounds is still open.  ...

### An upper bound on Euclidean embeddings of rigid graphs with 8 vertices [article]

Stylianos C. Despotakis, Ioannis Z. Emiris
2014 arXiv   pre-print
A graph is called (generically) rigid in R^d if, for any choice of sufficiently generic edge lengths, it can be embedded in R^d in a finite number of distinct ways, modulo rigid transformations.  ...  Here, we deal with the problem of determining the maximum number of planar Euclidean embeddings of minimally rigid graphs with 8 vertices, because this is the smallest unknown case in the plane.  ...  Our result is to prove an upper bound of 116 to the number of planar embeddings of minimally rigid graphs with 8 vertices in the plane. We conjecture that the bound of 116 is tight.  ...

### Page 2587 of Mathematical Reviews Vol. , Issue 97D [page]

1997 Mathematical Reviews
We also find tight lower and upper bounds for the j-linkage (equivalently, the j-width) of graphs with given numbers of vertices and edges.  ...  It is interesting to note that a lower bound for the width of a graph had been found by Erdés; as we show, however, that bound is not tight.  ...

### Irrelevant Vertices for the Planar Disjoint Paths Problem [article]

Isolde Adler, Stavros G. Kolliopoulos, Philipp Klaus Krause, Daniel Lokshtanov, Saket Saurabhh, Dimitrios M. Thilikos
2016 arXiv   pre-print
Our bound is radically better than the bounds known for general graphs.  ...  In this paper we give a new and self-contained proof of this result that strongly exploits the combinatorial properties of planar graphs and achieves g(k)=O(k^3/2· 2^k).  ...  We thank Ken-ichi Kawarabayashi and Paul Wollan for providing details on the bounds in  .  ...

### Irrelevant vertices for the planar Disjoint Paths Problem

Isolde Adler, Stavros G. Kolliopoulos, Philipp Klaus Krause, Daniel Lokshtanov, Saket Saurabh, Dimitrios M. Thilikos
2017 Journal of combinatorial theory. Series B (Print)
Our bound is radically better than the bounds known for general graphs.  ...  In this paper we give a new and self-contained proof of this result that strongly exploits the combinatorial properties of planar graphs and achieves g(k) = O(k 3 / 2 . 2 k ).  ...  This algorithm runs in 22O(k) • n steps. Acknowledgment. We thank Ken-ichi Kawarabayashi and Paul Wollan for providing details on the bounds in  .  ...

### Algebraic methods for counting Euclidean embeddings of rigid graphs [article]

Ioannis Z. Emiris, Elias P. Tsigaridas, Antonios Varvitsiotis
2009 arXiv   pre-print
Moreover, our implementation yields upper bounds for n < 10 in ^2 and ^3, which reduce the existing gaps, and tight bounds up to n=7 in ^3.  ...  The study of (minimally) rigid graphs is motivated by numerous applications, mostly in robotics and bioinformatics.  ...  V. thanks Günter Rote for insightful discussions on our conjecture. A.  ...

### Guest Editors' Foreword

Nikhil Bansal, Irene Finocchi
2017 Algorithmica
In "Multicuts in Planar and Bounded-Genus Graphs with Bounded Number of Terminals", de Verdiére presents an FPT algorithm based on topological techniques, for the minimum multicut problem on bounded-genus  ...  by norms in O(1) dimensions for the complete-linkage clustering algorithm, a popular method for computing hierarchical clusterings.  ...

### Combing a Linkage in an Annulus [article]

Petr A. Golovach and Giannos Stamoulis and Dimitrios M. Thilikos
2022 arXiv   pre-print
A linkage in a graph G of size k is a subgraph L of G whose connected components are k paths. The pattern of a linkage of size k is the set of k pairs formed by the endpoints of these paths.  ...  We deduce several variants of this result in the cases where s=0 and s>0. These variants permit the application of the unique linkage theorem on several path routing problems on embedded graphs.  ...  In fact, we prove Theorem 1 in a more general setting. The planarity condition is only necessary for the part of the graph bounded by the inner and outer cycle of C.  ...

### The Parameterized Complexity of Graph Cyclability

Petr A. Golovach, Marcin Kamiński, Spyridon Maniatis, Dimitrios M. Thilikos
2017 SIAM Journal on Discrete Mathematics
We give an FPT algorithm for planar graphs that runs in time 2 2 O(k 2 log k) · n 2 . Our algorithm is based on a series of graph theoretical results on cyclic linkages in planar graphs.  ...  The cyclability of a graph is the maximum integer k for which every k vertices lie on a cycle.  ...  Therefore S is cyclable in G, as required. ⊓ ⊔ Vital cyclic linkages Tight sequences.  ...

### The Parameterized Complexity of Graph Cyclability [chapter]

Petr A. Golovach, Marcin Kamiński, Spyridon Maniatis, Dimitrios M. Thilikos
2014 Lecture Notes in Computer Science
We give an FPT algorithm for planar graphs that runs in time 2 2 O(k 2 log k) · n 2 . Our algorithm is based on a series of graph theoretical results on cyclic linkages in planar graphs.  ...  The cyclability of a graph is the maximum integer k for which every k vertices lie on a cycle.  ...  Therefore S is cyclable in G, as required. ⊓ ⊔ Vital cyclic linkages Tight sequences.  ...

### The Parameterized Complexity of Graph Cyclability [article]

Petr A. Golovach, Marcin Kamiński, Spyridon Maniatis, Dimitrios M. Thilikos
2016 arXiv   pre-print
Our algorithm is based on a series of graph-theoretical results on cyclic linkages in planar graphs.  ...  On the positive side, we give an FPT algorithm for planar graphs that runs in time 2^2^O(k^2 k)· n^2.  ...  Kernelization lower bounds for planar graphs Now we show that it is unlikely that Cyclability, parameterized by k, has a polynomial kernel when restricted to planar graphs.  ...

### Order-preserving drawings of trees with approximately optimal height (and small width) [article]

Johannes Batzill, Therese Biedl
2016 arXiv   pre-print
Finally we construct trees that require height 2pw(T)+1 in all planar order-preserving straight-line drawings.  ...  In this paper, we study how to draw trees so that they are planar, straight-line and respect a given order of edges around each node.  ...  We also showed that '2' is tight if one uses the pathwidth for lower-bounding the height.  ...

### An Exponential Time Parameterized Algorithm for Planar Disjoint Paths [article]

Daniel Lokshtanov, Pranabendu Misra, Michal Pilipczuk, Saket Saurabh, Meirav Zehavi
2021 arXiv   pre-print
in single exponential time on surface-embedded graphs and in particular on planar graphs.  ...  In this paper, we make the first step towards our quest for designing a single exponential time algorithm for Disjoint Paths by giving a 2^O(k^2)n^O(1)-time algorithm for Planar Disjoint Paths.  ...   for Disjoint Paths on directed planar graphs.  ...

### Mixed Volume and Distance Geometry Techniques for Counting Euclidean Embeddings of Rigid Graphs [chapter]

Ioannis Z. Emiris, Elias P. Tsigaridas, Antonios Varvitsiotis
2012 Distance Geometry
Our implementation yields upper bounds for n ≤ 10 in R 2 and R 3 , which reduce the existing gaps and lead to tight bounds for n ≤ 7 in both R 2 and R 3 ; in particular, we describe the recent settlement  ...  Our approach also yields a new upper bound for Laman graphs with 8 vertices, which is conjectured to be tight.  ...  and the National Science Foundation of China (under grant 61061130540) for the Sino-Danish Center for the Theory of Interactive Computation, within which part of this work was performed, and from the  ...
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