An algorithm and upper bounds for the weighted maximal planar graph problem

2015 ◽  
Vol 66 (8) ◽  
pp. 1399-1412 ◽  
Author(s):  
Amir Ahmadi-Javid ◽  
Amir Ardestani-Jaafari ◽  
Leslie R Foulds ◽  
Hossein Hojabri ◽  
Reza Zanjirani Farahani
Keyword(s):  
Planar Graph ◽  
Upper Bounds ◽  
Maximal Planar Graph ◽  
Graph Problem

Facility Layout Planning

2013 ◽  
pp. 212-223
Author(s):  
Hossein Hojabri ◽  
Elnaz Miandoabchi
Keyword(s):  
Mathematical Models ◽  
Planar Graph ◽  
Facility Layout ◽  
Layout Design ◽  
Simple Graph ◽  
Layout Planning ◽  
Adjacency Graph ◽  
Manufacturing Plants ◽  
Maximal Planar Graph ◽  
Facilities Layout

The weighted maximal planar graph (WMPG) appears in many applications. It is currently used to design facilities layout in manufacturing plants. Given an edge-weighted complete simple graph G, the WMPG involves finding a sub-graph of G that is planar in the sense that it could be embedded on the plane such that none of its edges intersect, and is maximal in the sense that no more edges can be added to it unless its planarity is violated. Finally, it is optimal in the sense that the resulting maximal planar graph holds the maximum sum of edge weights. In this chapter, the aim is to explain the application of planarity in facility layout design. The mathematical models and the algorithms developed for the problem so far are explained. In the meanwhile, the corollaries and theorems needed to explain the algorithms and models are briefly given. In the last part, an explanation on how to draw block layout from the adjacency graph is given.

scholarly journals The maximum Wiener index of maximal planar graphs

2020 ◽  
Vol 40 (4) ◽  
pp. 1121-1135
Author(s):  
Debarun Ghosh ◽  
Ervin Győri ◽  
Addisu Paulos ◽  
Nika Salia ◽  
Oscar Zamora
Keyword(s):  
Planar Graph ◽  
Connected Graph ◽  
Planar Graphs ◽  
Wiener Index ◽  
Maximal Planar Graph

Abstract The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an n-vertex maximal planar graph is at most $$\lfloor \frac{1}{18}(n^3+3n^2)\rfloor $$ ⌊ 1 18 ( n 3 + 3 n 2 ) ⌋ . We prove this conjecture and determine the unique n-vertex maximal planar graph attaining this maximum, for every $$ n\ge 10$$ n ≥ 10 .

scholarly journals A 1-tough nonhamiltonian maximal planar graph

1980 ◽  
Vol 30 (3) ◽  
pp. 305-307 ◽  
Author(s):  
Takao Nishizeki
Keyword(s):  
Planar Graph ◽  
Maximal Planar Graph

scholarly journals Face-Degree Bounds for Planar Critical Graphs

10.37236/5895 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Ligang Jin ◽  
Yingli Kang ◽  
Eckhard Steffen
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Upper Bounds ◽  
Partial Characterization ◽  
Critical Graphs ◽  
Degree Bounds

The only remaining case of a well known conjecture of Vizing states that there is no planar graph with maximum degree 6 and edge chromatic number 7. We introduce parameters for planar graphs,  based on the degrees of the faces, and study the question whether there are upper bounds for these parameters for planar edge-chromatic critical graphs. Our results provide upper bounds on these parameters for smallest counterexamples to Vizing's conjecture, thus providing a partial characterization of such graphs, if they exist.For $k \leq 5$ the results give insights into the structure of planar edge-chromatic critical graphs.

On the number of cycles of lengthk in a maximal planar graph

1979 ◽  
Vol 3 (1) ◽  
pp. 69-86 ◽  
Author(s):  
S. L. Hakimi ◽  
E. F. Schmeichel
Keyword(s):  
Planar Graph ◽  
Maximal Planar Graph ◽  
Number Of Cycles
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