string graphs
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Author(s):  
István Tomon

AbstractA string graph is the intersection graph of curves in the plane. We prove that for every $$\epsilon >0$$ ϵ > 0 , if G is a string graph with n vertices such that the edge density of G is below $${1}/{4}-\epsilon $$ 1 / 4 - ϵ , then V(G) contains two linear sized subsets A and B with no edges between them. The constant 1/4 is a sharp threshold for this phenomenon as there are string graphs with edge density less than $${1}/{4}+\epsilon $$ 1 / 4 + ϵ such that there is an edge connecting any two logarithmic sized subsets of the vertices. The existence of linear sized sets A and B with no edges between them in sufficiently sparse string graphs is a direct consequence of a recent result of Lee about separators. Our main theorem finds the largest possible density for which this still holds. In the special case when the curves are x-monotone, the same result was proved by Pach and the author of this paper, who also proposed the conjecture for the general case.


2021 ◽  
pp. 593-598
Author(s):  
Sandip Das ◽  
Joydeep Mukherjee ◽  
Uma kant Sahoo
Keyword(s):  

2020 ◽  
Vol 63 (4) ◽  
pp. 888-917
Author(s):  
János Pach ◽  
Bruce Reed ◽  
Yelena Yuditsky

Author(s):  
Chaya Keller ◽  
Alexandre Rok ◽  
Shakhar Smorodinsky
Keyword(s):  

Author(s):  
Madhumangal Pal

In this chapter, a very important class of graphs called intersection graph is introduced. Based on the geometrical representation, many different types of intersection graphs can be defined with interesting properties. Some of them—interval graphs, circular-arc graphs, permutation graphs, trapezoid graphs, chordal graphs, line graphs, disk graphs, string graphs—are presented here. A brief introduction of each of these intersection graphs along with some basic properties and algorithmic status are investigated.


10.37236/8096 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Vít Jelínek ◽  
Martin Töpfer

We consider the graph classes Grounded-L and Grounded-{𝖫,⅃} corresponding to graphs that admit an intersection representation by 𝖫-shaped curves (or 𝖫-shaped and ⅃-shaped curves, respectively), where additionally the topmost points of each curve are assumed to belong to a common horizontal line. We prove that Grounded-L graphs admit an equivalent characterisation in terms of vertex ordering with forbidden patterns. We also compare these classes to related intersection classes, such as the grounded segment graphs, the monotone 𝖫-graphs (a.k.a. max point-tolerance graphs), or the outer-1-string graphs. We give constructions showing that these classes are all distinct and satisfy only trivial or previously known inclusions.


Algorithmica ◽  
2019 ◽  
Vol 81 (7) ◽  
pp. 3047-3073 ◽  
Author(s):  
Édouard Bonnet ◽  
Paweł Rzążewski
Keyword(s):  

2019 ◽  
Vol 35 (19) ◽  
pp. 3599-3607 ◽  
Author(s):  
Mikko Rautiainen ◽  
Veli Mäkinen ◽  
Tobias Marschall

Abstract Motivation Graphs are commonly used to represent sets of sequences. Either edges or nodes can be labeled by sequences, so that each path in the graph spells a concatenated sequence. Examples include graphs to represent genome assemblies, such as string graphs and de Bruijn graphs, and graphs to represent a pan-genome and hence the genetic variation present in a population. Being able to align sequencing reads to such graphs is a key step for many analyses and its applications include genome assembly, read error correction and variant calling with respect to a variation graph. Results We generalize two linear sequence-to-sequence algorithms to graphs: the Shift-And algorithm for exact matching and Myers’ bitvector algorithm for semi-global alignment. These linear algorithms are both based on processing w sequence characters with a constant number of operations, where w is the word size of the machine (commonly 64), and achieve a speedup of up to w over naive algorithms. For a graph with |V| nodes and |E| edges and a sequence of length m, our bitvector-based graph alignment algorithm reaches a worst case runtime of O(|V|+⌈mw⌉|E| log w) for acyclic graphs and O(|V|+m|E| log w) for arbitrary cyclic graphs. We apply it to five different types of graphs and observe a speedup between 3-fold and 20-fold compared with a previous (asymptotically optimal) alignment algorithm. Availability and implementation https://github.com/maickrau/GraphAligner Supplementary information Supplementary data are available at Bioinformatics online.


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