scholarly journals Outerplanar Crossing Numbers, the Circular Arrangement Problem and Isoperimetric Functions

10.37236/1834 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Éva Czabarka ◽  
Ondrej Sýkora ◽  
László A. Székely ◽  
Imrich Vrťo
Keyword(s):  
Lower Bound ◽  
General Setting ◽  
Crossing Number ◽  
Linear Arrangement ◽  
Crossing Numbers ◽  
Arrangement Problem ◽  
Isoperimetric Functions ◽  
Circular Arrangement ◽  
Graph Families ◽  
Linear Arrangement Problem

We extend a lower bound due to Shahrokhi, Sýkora, Székely and Vrťo for the outerplanar crossing number (in other terminologies also called convex, circular and one-page book crossing number) to a more general setting. In this setting we can show a better lower bound for the outerplanar crossing number of hypercubes than the best lower bound for the planar crossing number. We exhibit further sequences of graphs, whose outerplanar crossing number exceeds by a factor of $\log n$ the planar crossing number of the graph. We study the circular arrangement problem, as a lower bound for the linear arrangement problem, in a general fashion. We obtain new lower bounds for the circular arrangement problem. All the results depend on establishing good isoperimetric functions for certain classes of graphs. For several graph families new near-tight isoperimetric functions are established.

Lower and upper bounds for the linear arrangement problem on interval graphs

2018 ◽  
Vol 52 (4-5) ◽  
pp. 1123-1145
Author(s):  
Alain Quilliot ◽  
Djamal Rebaine ◽  
Hélène Toussaint
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Feasible Solution ◽  
Interval Graph ◽  
Upper Bounds ◽  
Unit Interval ◽  
Lower And Upper Bounds ◽  
Linear Arrangement ◽  
Arrangement Problem ◽  
Linear Arrangement Problem

We deal here with theLinear Arrangement Problem(LAP) onintervalgraphs, any interval graph being given here together with its representation as theintersectiongraph of some collection of intervals, and so with relatedprecedenceandinclusionrelations. We first propose a lower boundLB, which happens to be tight in the case ofunit intervalgraphs. Next, we introduce the restriction PCLAP of LAP which is obtained by requiring any feasible solution of LAP to be consistent with theprecedencerelation, and prove that PCLAP can be solved in polynomial time. Finally, we show both theoretically and experimentally that PCLAP solutions are a good approximation for LAP onintervalgraphs.

scholarly journals Braid index bounds ropelength from below

2020 ◽  
Vol 29 (04) ◽  
pp. 2050019
Author(s):  
Yuanan Diao
Keyword(s):  
Lower Bound ◽  
General Formula ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Braid Index ◽  
Alternating Links

For an unoriented link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text]. It is known that in general [Formula: see text] is at least of the order [Formula: see text], and at most of the order [Formula: see text] where [Formula: see text] is the minimum crossing number of [Formula: see text]. Furthermore, it is known that there exist families of (infinitely many) links with the property [Formula: see text]. A long standing open conjecture states that if [Formula: see text] is alternating, then [Formula: see text] is at least of the order [Formula: see text]. In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of [Formula: see text] (called the maximum braid index of [Formula: see text]). Consequently, [Formula: see text] for any link [Formula: see text] whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links.

scholarly journals Tractable Parameterizations for the Minimum Linear Arrangement Problem

2016 ◽  
Vol 8 (2) ◽  
pp. 1-12 ◽  
Author(s):  
Michael R. Fellows ◽  
Danny Hermelin ◽  
Frances Rosamond ◽  
Hadas Shachnai
Keyword(s):  
Linear Arrangement ◽  
Arrangement Problem ◽  
Linear Arrangement Problem

scholarly journals On minimum cuts and the linear arrangement problem

2000 ◽  
Vol 103 (1-3) ◽  
pp. 127-139 ◽  
Author(s):  
S.B. Horton ◽  
R.Gary Parker ◽  
R.B. Borie
Keyword(s):  
Linear Arrangement ◽  
Minimum Cuts ◽  
Arrangement Problem ◽  
Linear Arrangement Problem

On Bipartite Drawings and the Linear Arrangement Problem

2001 ◽  
Vol 30 (6) ◽  
pp. 1773-1789 ◽  
Author(s):  
Farhad Shahrokhi ◽  
Ondrej Sýkora ◽  
László A. Székely ◽  
Imrich Vrto
Keyword(s):  
Linear Arrangement ◽  
Arrangement Problem ◽  
Linear Arrangement Problem

scholarly journals Crossing Numbers and Hard Erdős Problems in Discrete Geometry

1997 ◽  
Vol 6 (3) ◽  
pp. 353-358 ◽  
Author(s):  
LÁSZLÓ A. SZÉKELY
Keyword(s):  
Lower Bound ◽  
Discrete Geometry ◽  
Crossing Number ◽  
Plane Geometry ◽  
Crossing Numbers ◽  
Minimum Number

We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.

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