scholarly journals The open-locating-dominating number of some convex polytopes

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 635-642
Author(s):  
Aleksandar Savic ◽  
Zoran Maksimovic ◽  
Milena Bogdanovic
Keyword(s):  
Planar Graphs ◽  
Convex Polytopes ◽  
Upper Bounds ◽  
Dominating Number

In this paper we will investigate the problem of finding the open-locating-dominating number for some classes of planar graphs - convex polytopes. We considered Dn, Tn, Bn, Cn, En and Rn classes of convex polytopes known from the literature. The exact values of open-locating-dominating number for Dn and Rn polytopes are presented, along with the upper bounds for Tn, Bn, Cn, and En polytopes.

scholarly journals Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 727 ◽  
Author(s):  
Hassan Raza ◽  
Sakander Hayat ◽  
Xiang-Feng Pan
Keyword(s):  
Convex Polytopes ◽  
Convex Polytope ◽  
Upper Bounds ◽  
Symmetric Convex ◽  
Dominating Sets ◽  
Incidence Relation ◽  
Dominating Number ◽  
Rotationally Symmetric ◽  
Finite Set ◽  
Set Of Points

A convex polytope or simply polytope is the convex hull of a finite set of points in Euclidean space R d . Graphs of convex polytopes emerge from geometric structures of convex polytopes by preserving the adjacency-incidence relation between vertices. In this paper, we study the problem of binary locating-dominating number for the graphs of convex polytopes which are symmetric rotationally. We provide an integer linear programming (ILP) formulation for the binary locating-dominating problem of graphs. We have determined the exact values of the binary locating-dominating number for two infinite families of convex polytopes. The exact values of the binary locating-dominating number are obtained for two rotationally-symmetric convex polytopes families. Moreover, certain upper bounds are determined for other three infinite families of convex polytopes. By using the ILP formulation, we show tightness in the obtained upper bounds.

New upper bounds on linear coloring of planar graphs

2011 ◽  
Vol 28 (6) ◽  
pp. 1187-1196 ◽  
Author(s):  
Bin Liu ◽  
Gui Zhen Liu
Keyword(s):  
Planar Graphs ◽  
Upper Bounds

scholarly journals Some Bounds for the Kirchhoff Index of Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yujun Yang
Keyword(s):  
Connected Graph ◽  
Planar Graphs ◽  
Independence Number ◽  
Clique Number ◽  
Upper Bounds ◽  
Electrical Network ◽  
Lower And Upper Bounds ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Fullerene Graphs

The resistance distance between two vertices of a connected graphGis defined as the effective resistance between them in the corresponding electrical network constructed fromGby replacing each edge ofGwith a unit resistor. The Kirchhoff index ofGis the sum of resistance distances between all pairs of vertices. In this paper, general bounds for the Kirchhoff index are given via the independence number and the clique number, respectively. Moreover, lower and upper bounds for the Kirchhoff index of planar graphs and fullerene graphs are investigated.

scholarly journals Counting Triangulations of Planar Point Sets

10.37236/557 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Micha Sharir ◽  
Adam Sheffer
Keyword(s):  
Upper Bound ◽  
Planar Graphs ◽  
Spanning Trees ◽  
Upper Bounds ◽  
Line Graphs ◽  
Point Sets ◽  
Planar Point ◽  
Straight Line ◽  
Point Set ◽  
Spanning Cycles

We study the maximal number of triangulations that a planar set of $n$ points can have, and show that it is at most $30^n$. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to the previous best upper bound of $43^n$ for the problem. Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossing-free) straight-line graphs on a given point set. Specifically, it can be used to derive new upper bounds for the number of planar graphs ($207.84^n$), spanning cycles ($O(68.67^n)$), spanning trees ($O(146.69^n)$), and cycle-free graphs ($O(164.17^n)$).

scholarly journals Counting Plane Graphs: Cross-Graph Charging Schemes

2013 ◽  
Vol 22 (6) ◽  
pp. 935-954 ◽  
Author(s):  
MICHA SHARIR ◽  
ADAM SHEFFER
Keyword(s):  
Upper Bound ◽  
Planar Graphs ◽  
Upper Bounds ◽  
Straight Edge ◽  
Specific Point ◽  
Plane Graphs ◽  
Fixed Set ◽  
Point Set ◽  
Vertex Degrees ◽  
Set Of Points

We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have recently been used to obtain various properties of triangulations that are embedded in a fixed set of points in the plane. We generalize this method to obtain results for various other types of graphs that are embedded in the plane. Specifically, we obtain a new bound ofO*(187.53N) (where theO*(⋅) notation hides polynomial factors) for the maximum number of crossing-free straight-edge graphs that can be embedded in any specific set ofNpoints in the plane (improving upon the previous best upper bound 207.85Nin Hoffmann, Schulz, Sharir, Sheffer, Tóth and Welzl [14]). We also derive upper bounds for numbers of several other types of plane graphs (such as connected and bi-connected plane graphs), and obtain various bounds on the expected vertex-degrees in graphs that are uniformly chosen from the set of all crossing-free straight-edge graphs that can be embedded in a specific point set.We then apply the cross-graph charging-scheme method to graphs that allow certain types of crossings. Specifically, we consider graphs with no set ofkpairwise crossing edges (more commonly known ask-quasi-planar graphs). Fork=3 andk=4, we prove that, for any setSofNpoints in the plane, the number of graphs that have a straight-edgek-quasi-planar embedding overSis only exponential inN.

New upper bounds on the decomposability of planar graphs

2005 ◽  
Vol 51 (1) ◽  
pp. 53-81 ◽  
Author(s):  
Fedor V. Fomin ◽  
Dimitrios M. Thilikos
Keyword(s):  
Planar Graphs ◽  
Upper Bounds

scholarly journals The Roman domination number of some special classes of graphs - convex polytopes

2021 ◽  
pp. 19-19
Author(s):  
Aleksandar Kartelj ◽  
Milana Grbic ◽  
Dragan Matic ◽  
Vladimir Filipovic
Keyword(s):  
Lower Bounds ◽  
Planar Graphs ◽  
Convex Polytopes ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Special Classes

In this paper we study the Roman domination number of some classes of planar graphs - convex polytopes: An, Rn and Tn. We establish the exact values of Roman domination number for: An, R3k, R3k+1, T8k, T8k+2, T8k+3, T8k+5 and T8k+6. For R3k+2, T8k+1, T8k+4 and T8k-1 we propose new upper and lower bounds, proving that the gap between the bounds is 1 for all cases except for the case of T8k+4, where the gap is 2.

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