scholarly journals Near-Optimal Separators in String Graphs

2013 ◽  
Vol 23 (1) ◽  
pp. 135-139 ◽  
Author(s):  
JIŘÍ MATOUŠEK
Keyword(s):  
Intersection Graph ◽  
String Graphs ◽  
String Graph

Let G be a string graph (an intersection graph of continuous arcs in the plane) with m edges. Fox and Pach proved that G has a separator consisting of $O(m^{3/4}\sqrt{\log m})$ vertices, and they conjectured that the bound of $O(\sqrt m)$ actually holds. We obtain separators with $O(\sqrt m \,\log m)$ vertices.

scholarly journals A Separator Theorem for String Graphs and its Applications

2009 ◽  
Vol 19 (3) ◽  
pp. 371-390 ◽  
Author(s):  
JACOB FOX ◽  
JÁNOS PACH
Keyword(s):  
Intersection Graph ◽  
Equal Size ◽  
Bipartite Subgraph ◽  
String Graphs ◽  
String Graph ◽  
Complete Bipartite Subgraph ◽  
Labelled Trees ◽  
Separator Theorem ◽  
Labelled Graphs

A string graph is the intersection graph of a collection of continuous arcs in the plane. We show that any string graph with m edges can be separated into two parts of roughly equal size by the removal of $O(m^{3/4}\sqrt{\log m})$ vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph Kt,t has at most ctn edges, where ct is a constant depending only on t. Another application shows that locally tree-like string graphs are globally tree-like: for any ε > 0, there is an integer g(ε) such that every string graph with n vertices and girth at least g(ε) has at most (1 + ε)n edges. Furthermore, the number of such labelled graphs is at most (1 + ε)nT(n), where T(n) = nn−2 is the number of labelled trees on n vertices.

scholarly journals Applications of a New Separator Theorem for String Graphs

2013 ◽  
Vol 23 (1) ◽  
pp. 66-74 ◽  
Author(s):  
JACOB FOX ◽  
JÁNOS PACH
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Present Note ◽  
Intersection Graph ◽  
Type Result ◽  
Vertex Separator ◽  
String Graphs ◽  
String Graph ◽  
Ramsey Type ◽  
Separator Theorem

An intersection graph of curves in the plane is called astring graph. Matoušek almost completely settled a conjecture of the authors by showing that every string graph withmedges admits a vertex separator of size$O(\sqrt{m}\log m)$. In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphsGwithnvertices: (i) ifKt⊈Gfor somet, then the chromatic number ofGis at most (logn)O(logt); (ii) ifKt,t⊈G, thenGhas at mostt(logt)O(1)nedges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdős–Hajnal conjecture almost holds for string graphs.

scholarly journals A Sharp Threshold Phenomenon in String Graphs

2021 ◽  
Author(s):  
István Tomon
Keyword(s):  
Recent Result ◽  
Direct Consequence ◽  
Intersection Graph ◽  
Edge Density ◽  
Sharp Threshold ◽  
Threshold Phenomenon ◽  
String Graphs ◽  
String Graph ◽  
Special Case

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.

An Introduction to Intersection Graphs

2020 ◽  
pp. 24-65 ◽  
Author(s):  
Madhumangal Pal
Keyword(s):  
Interval Graphs ◽  
Chordal Graphs ◽  
Intersection Graph ◽  
Important Class ◽  
Intersection Graphs ◽  
Permutation Graphs ◽  
Circular Arc ◽  
String Graphs ◽  
Circular Arc Graphs ◽  
Trapezoid Graphs

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.

scholarly journals Study on the Identification Method of Pyrogenetic Rock Based on N -DP Intersection Graph

2021 ◽  
Vol 804 (2) ◽  
pp. 022020
Author(s):  
Gang Hu ◽  
Zhiqiang Mao ◽  
Ping Yan ◽  
Qian Yin ◽  
Xiyu Wang ◽  
...  
Keyword(s):  
Intersection Graph ◽  
Identification Method

scholarly journals Turán-type results for intersection graphs of boxes

2021 ◽  
pp. 1-6
Author(s):  
István Tomon ◽  
Dmitriy Zakharov
Keyword(s):  
Short Note ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Poset Dimension ◽  
Axis Parallel ◽  
Turán Theorem ◽  
Separation Dimension

Abstract In this short note, we prove the following analog of the Kővári–Sós–Turán theorem for intersection graphs of boxes. If G is the intersection graph of n axis-parallel boxes in $${{\mathbb{R}}^d}$$ such that G contains no copy of K t,t , then G has at most ctn( log n)2d+3 edges, where c = c(d)>0 only depends on d. Our proof is based on exploring connections between boxicity, separation dimension and poset dimension. Using this approach, we also show that a construction of Basit, Chernikov, Starchenko, Tao and Tran of K2,2-free incidence graphs of points and rectangles in the plane can be used to disprove a conjecture of Alon, Basavaraju, Chandran, Mathew and Rajendraprasad. We show that there exist graphs of separation dimension 4 having superlinear number of edges.

scholarly journals The intersection graph of a finite simple group has diameter at most 5

2021 ◽  
Author(s):  
Saul D. Freedman
Keyword(s):  
Simple Group ◽  
Upper Bound ◽  
Finite Simple Group ◽  
Unitary Groups ◽  
Intersection Graph ◽  
Monster Group ◽  
Prime Dimension

AbstractLet G be a non-abelian finite simple group. In addition, let $$\Delta _G$$ Δ G be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect non-trivially. We prove that the diameter of $$\Delta _G$$ Δ G has a tight upper bound of 5, thereby resolving a question posed by Shen (Czechoslov Math J 60(4):945–950, 2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.

A Krasnosel’skii-type theorem for certain orthogonal polytopes starshaped via k-staircase paths

2020 ◽  
Vol 20 (2) ◽  
pp. 169-177 ◽  
Author(s):  
Marilyn Breen
Keyword(s):  
Type Theorem ◽  
Common Point ◽  
Finite Family ◽  
Intersection Graph ◽  
Axis Parallel ◽  
Orthogonal Polytopes

AbstractLet 𝒞 be a finite family of distinct axis-parallel boxes in ℝd whose intersection graph is a tree, and let S = ⋃{C : C in 𝒞}. If every two points of S see a common point of S via k-staircase paths, then S is starshaped via k-staircase paths. Moreover, the k-staircase kernel of S will be convex via k-staircases.

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