scholarly journals On Almost-Planar Graphs

10.37236/6591 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Guoli Ding ◽  
Joshua Fallon ◽  
Emily Marshall
Keyword(s):  
Planar Graphs ◽  
Short Proof ◽  
Nonplanar Graph

A nonplanar graph $G$ is called almost-planar if for every edge $e$ of $G$, at least one of $G\backslash e$ and $G/e$ is planar. In 1990, Gubser characterized 3-connected almost-planar graphs in his dissertation. However, his proof is so long that only a small portion of it was published. The main purpose of this paper is to provide a short proof of this result. We also discuss the structure of almost-planar graphs that are not 3-connected.

scholarly journals KNOT-INEVITABLE PROJECTIONS OF PLANAR GRAPHS

1996 ◽  
Vol 05 (06) ◽  
pp. 877-883 ◽  
Author(s):  
KOUKI TANIYAMA ◽  
TATSUYA TSUKAMOTO
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Spatial Graph ◽  
Torus Knot ◽  
Nonplanar Graph

For each odd number n, we describe a regular projection of a planar graph such that every spatial graph obtained by giving it over/under information of crossing points contains a (2, n)-torus knot. We also show that for any spatial graph H, there is a regular projection of a (possibly nonplanar) graph such that every spatial graph obtained from it contains a subgraph that is ambient isotopic to H.

scholarly journals Planar graphs with maximum degree Δ≥9 are (Δ+1)-edge-choosable—A short proof

2010 ◽  
Vol 310 (21) ◽  
pp. 3049-3051 ◽  
Author(s):  
Nathann Cohen ◽  
Frédéric Havet
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Short Proof

scholarly journals Graphs with no K3,3 Minor Containing a Fixed Edge

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Donald K. Wagner
Keyword(s):  
Efficient Algorithm ◽  
General Class ◽  
Planar Graphs ◽  
Flow Problem ◽  
Maximum Flow ◽  
Maximum Flow Problem ◽  
Minimal Cut ◽  
Nonplanar Graph ◽  
The Given

It is well known that every cycle of a graph must intersect every cut in an even number of edges. For planar graphs, Ford and Fulkerson proved that, for any edge e, there exists a cycle containing e that intersects every minimal cut containing e in exactly two edges. The main result of this paper generalizes this result to any nonplanar graph G provided G does not have a K3,3 minor containing the given edge e. Ford and Fulkerson used their result to provide an efficient algorithm for solving the maximum-flow problem on planar graphs. As a corollary to the main result of this paper, it is shown that the Ford-Fulkerson algorithm naturally extends to this more general class of graphs.

scholarly journals Coloring with no $2$-Colored $P_4$'s

10.37236/1779 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Michael O. Albertson ◽  
Glenn G. Chappell ◽  
H. A. Kierstead ◽  
André Kündgen ◽  
Radhika Ramamurthi
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Short Proof ◽  
Cubic Graphs ◽  
Proper Coloring ◽  
Be Star ◽  
Tree Width ◽  
Star Coloring

A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that every acyclic $k$-coloring can be refined to a star coloring with at most $(2k^2-k)$ colors. Similarly, we prove that planar graphs have star colorings with at most 20 colors and we exhibit a planar graph which requires 10 colors. We prove several other structural and topological results for star colorings, such as: cubic graphs are $7$-colorable, and planar graphs of girth at least $7$ are $9$-colorable. We provide a short proof of the result of Fertin, Raspaud, and Reed that graphs with tree-width $t$ can be star colored with ${t+2\choose2}$ colors, and we show that this is best possible.

scholarly journals Clustered Planarity Testing Revisited

10.37236/5002 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Radoslav Fulek ◽  
Jan Kynčl ◽  
Igor Malinović ◽  
Dömötör Pálvölgyi
Keyword(s):  
Polynomial Time ◽  
Planar Graphs ◽  
Classical Result ◽  
Short Proof ◽  
Matroid Intersection ◽  
Intersection Algorithm ◽  
Clustered Planarity ◽  
First Time ◽  
Straightforward Extension

The Hanani–Tutte theorem is a classical result proved for the first time in the 1930s that characterizes planar graphs as graphs that admit a drawing in the plane in which every pair of edges not sharing a vertex cross an even number of times. We generalize this result to clustered graphs with two disjoint clusters, and show that a straightforward extension to flat clustered graphs with three or more disjoint clusters is not possible. For general clustered graphs we show a variant of the Hanani–Tutte theorem in the case when each cluster induces a connected subgraph.Di Battista and Frati proved that clustered planarity of embedded clustered graphs whose every face is incident with at most five vertices can be tested in polynomial time. We give a new and short proof of this result, using the matroid intersection algorithm.

Acute Constraints in Straight-Line Drawings of Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 994-1001
Author(s):  
Akane SETO ◽  
Aleksandar SHURBEVSKI ◽  
Hiroshi NAGAMOCHI ◽  
Peter EADES
Keyword(s):  
Planar Graphs ◽  
Line Drawings ◽  
Straight Line

A Space-Efficient Separator Algorithm for Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 1007-1016 ◽  
Author(s):  
Ryo ASHIDA ◽  
Sebastian KUHNERT ◽  
Osamu WATANABE
Keyword(s):  
Planar Graphs

A note on generalized quasi-contraction

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3091-3093
Author(s):  
Dejan Ilic ◽  
Darko Kocev
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Short Proof

In this paper we give a short proof of the main results of Kumam, Dung and Sitthithakerngkiet (P. Kumam, N.V. Dung, K. Sitthithakerngkiet, A Generalization of Ciric Fixed Point Theorems, FILOMAT 29:7 (2015), 1549-1556).

Short proof that Kneser graphs are Hamiltonian for n ⩾ 4k

2021 ◽  
Vol 344 (7) ◽  
pp. 112430
Author(s):  
Johann Bellmann ◽  
Bjarne Schülke
Keyword(s):  
Short Proof ◽  
Kneser Graphs

scholarly journals Strongly perfect claw‐free graphs—A short proof

2021 ◽  
Author(s):  
Maria Chudnovsky ◽  
Cemil Dibek
Keyword(s):  
Short Proof
Sign in / Sign up

Export Citation Format

Share Document