scholarly journals Non-Separating Planar Graphs

10.37236/8816 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Hooman R. Dehkordi ◽  
Graham Farr
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Disjoint Union ◽  
Disjoint Paths ◽  
Outerplanar Graph ◽  
Outerplanar Graphs ◽  
Structural Characterisation ◽  
A Minor ◽  
Vertex Disjoint

A graph $G$ is a non-separating planar graph if there is a drawing $D$ of $G$ on the plane such that (1) no two edges cross each other in $D$ and (2) for any cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in $D$. Non-separating planar graphs are closed under taking minors and are a subclass of planar graphs and a superclass of outerplanar graphs. In this paper, we show that a graph is a non-separating planar graph if and only if it does not contain $K_1 \cup K_4$ or $K_1 \cup K_{2,3}$ or $K_{1,1,3}$ as a minor. Furthermore, we provide a structural characterisation of this class of graphs. More specifically, we show that any maximal non-separating planar graph is either an outerplanar graph or a wheel or it is a graph obtained from the disjoint union of two triangles by adding three vertex-disjoint paths between the two triangles. Lastly, to demonstrate an application of non-separating planar graphs, we use the characterisation of non-separating planar graphs to prove that there are maximal linkless graphs with $3n-3$ edges. Thus, maximal linkless graphs can have significantly fewer edges than maximum linkless graphs; Sachs exhibited linkless graphs with $n$ vertices and $4n-10$ edges (the maximum possible) in 1983.

scholarly journals Path Choosability of Planar Graphs

10.37236/7139 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Glenn G. Chappell ◽  
Chris Hartman
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Disjoint Union ◽  
Vertex Coloring ◽  
List Coloring ◽  
Outerplanar Graphs ◽  
Color Class ◽  
Path Coloring

A path coloring of a graph $G$ is a vertex coloring of $G$ such that each color class induces a disjoint union of paths. We consider a path-coloring version of list coloring for planar and outerplanar graphs. We show that if each vertex of a planar graph is assigned a list of $3$ colors, then the graph admits a path coloring in which each vertex receives a color from its list. We prove a similar result for outerplanar graphs and lists of size $2$.For outerplanar graphs we prove a multicoloring generalization. We assign each vertex of a graph a list of $q$ colors. We wish to color each vertex with $r$ colors from its list so that, for each color, the set of vertices receiving it induces a disjoint union of paths. We show that we can do this for all outerplanar graphs if and only if $q/r \ge 2$. For planar graphs we conjecture that a similar result holds with $q/r \ge 3$; we present partial results toward this conjecture.

CONSTRAINED POINT-SET EMBEDDABILITY OF PLANAR GRAPHS

2010 ◽  
Vol 20 (05) ◽  
pp. 577-600 ◽  
Author(s):  
EMILIO DI GIACOMO ◽  
WALTER DIDIMO ◽  
GIUSEPPE LIOTTA ◽  
HENK MEIJER ◽  
STEPHEN K. WISMATH
Keyword(s):  
Planar Graph ◽  
General Position ◽  
Planar Graphs ◽  
Disjoint Paths ◽  
Simple Path ◽  
Outerplanar Graph ◽  
Straight Line ◽  
Point Set ◽  
Number Of Bends ◽  
Set Of Points

This paper starts the investigation of a constrained version of the point-set embed-dability problem. Let G = (V,E) be a planar graph with n vertices, G′ = (V′,E′) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G′ is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that: • If G′ is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E\E′ have at most 4 bends each. • If S is any set of points in general position and G′ is a face of G or if it is a simple path, the curve complexity of the edges of E\E′ is at most 8. • If S is in general position and G′ is a set of k disjoint paths, the curve complexity of the edges of E \ E′ is O(2k).

scholarly journals Planar Graphs have Independence Ratio at least 3/13

10.37236/5309 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Daniel W. Cranston ◽  
Landon Rabern
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Disjoint Union ◽  
Independence Number ◽  
Independent Set

The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an independent set of size at least $\frac{n}4$; this is best possible, as shown by the disjoint union of many copies of $K_4$.  In 1968, Erdős asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex planar graph has an independent set of size at least $\frac{2n}9$. Until now, this remained the best bound independent of the 4CT. Our main result improves this bound to $\frac{3n}{13}$.

Gossiping in vertex-disjoint paths mode in d-dimensional grids and planar graphs

1993 ◽  
pp. 200-211 ◽  
Author(s):  
Juraj Hromkovič ◽  
Ralf Klasing ◽  
Elena A. Stöhr ◽  
Hubert Wagener
Keyword(s):  
Planar Graphs ◽  
Disjoint Paths ◽  
Vertex Disjoint

scholarly journals Gossiping in Vertex-Disjoint Paths Mode in d-Dimensional Grids and Planar Graphs

1995 ◽  
Vol 123 (1) ◽  
pp. 17-28 ◽  
Author(s):  
J. Hromkovic ◽  
R. Klasing ◽  
E.A. Stohr ◽  
H. Wagener
Keyword(s):  
Planar Graphs ◽  
Disjoint Paths ◽  
Vertex Disjoint

LINEAR-TIME ALGORITHMS FOR DISJOINT TWO-FACE PATHS PROBLEMS IN PLANAR GRAPHS

1996 ◽  
Vol 07 (02) ◽  
pp. 95-110 ◽  
Author(s):  
HEIKE RIPPHAUSEN-LIPA ◽  
DOROTHEA WAGNER ◽  
KARSTEN WEIHE
Keyword(s):  
Planar Graphs ◽  
Linear Time ◽  
Time Algorithm ◽  
Disjoint Paths ◽  
Linear Time Algorithm ◽  
Vertex Disjoint ◽  
Linear Time Algorithms

In this paper we present a linear-time algorithm for the vertex-disjoint Two-Face Paths Problem in planar graphs, i.e., the problem of finding k vertex-disjoint paths between pairs of terminals which lie on two face boundaries. The algorithm is based on the idea of finding rightmost paths with a certain property in planar graphs. Using this method, a linear-time algorithm for finding vertex-disjoint paths of a prescribed homotopy is derived. Moreover, the algorithm is modified to solve the more general linkage problem in linear time, as well.

scholarly journals Vertex Disjoint Paths in Upward Planar Graphs

2014 ◽  
pp. 52-64 ◽  
Author(s):  
Saeed Akhoondian Amiri ◽  
Ali Golshani ◽  
Stephan Kreutzer ◽  
Sebastian Siebertz
Keyword(s):  
Planar Graphs ◽  
Disjoint Paths ◽  
Vertex Disjoint

On square-free vertex colorings of graphs

2007 ◽  
Vol 44 (3) ◽  
pp. 411-422 ◽  
Author(s):  
János Barát ◽  
Péter Varjú
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Path Following ◽  
Fixed Number ◽  
The Other ◽  
Graph Colorings ◽  
Outerplanar Graphs ◽  
Free Vertex ◽  
Other Hand ◽  
Color Sequence

A sequence of symbols a1 , a2 … is called square-free if it does not contain a subsequence of consecutive terms of the form x1 , …, xm , x1 , …, xm . A century ago Thue showed that there exist arbitrarily long square-free sequences using only three symbols. Sequences can be thought of as colors on the vertices of a path. Following the paper of Alon, Grytczuk, Hałuszczak and Riordan, we examine graph colorings for which the color sequence is square-free on any path. The main result is that the vertices of any k -tree have a coloring of this kind using O ( ck ) colors if c > 6. Alon et al. conjectured that a fixed number of colors suffices for any planar graph. We support this conjecture by showing that this number is at most 12 for outerplanar graphs. On the other hand we prove that some outerplanar graphs require at least 7 colors. Using this latter we construct planar graphs, for which at least 10 colors are necessary.

SIMULTANEOUS EMBEDDING OF OUTERPLANAR GRAPHS, PATHS, AND CYCLES

2007 ◽  
Vol 17 (02) ◽  
pp. 139-160 ◽  
Author(s):  
EMILIO DI GIACOMO ◽  
GIUSEPPE LIOTTA
Keyword(s):  
Planar Graphs ◽  
The Other ◽  
Simple Path ◽  
Outerplanar Graph ◽  
Outerplanar Graphs ◽  
Line Segments ◽  
Straight Line ◽  
Simultaneous Embedding

Let G1 and G2 be two planar graphs having some vertices in common. A simultaneous embedding of G1 and G2 is a pair of crossing-free drawings of G1 and G2 such that each vertex in common is represented by the same point in both drawings. In this paper we show that an outerplanar graph and a simple path can be simultaneously embedded with fixed edges such that the edges in common are straight-line segments while the other edges of the outerplanar graph can have at most one bend per edge. We then exploit the technique for outerplanar graphs and paths to study simultaneous embeddings of other pairs of graphs. Namely, we study simultaneous embedding with fixed edges of: (i) two outerplanar graphs sharing a forest of paths and (ii) an outerplanar graph and a cycle.

List Improper Colourings of Planar Graphs

1999 ◽  
Vol 8 (3) ◽  
pp. 293-299 ◽  
Author(s):  
R. šKREKOVSKI
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Outerplanar Graph

A graph G is m-choosable with impropriety d, or simply (m, d)*-choosable, if for every list assignment L, where [mid ]L(v)[mid ][ges ]m for every v∈V(G), there exists an L-colouring of G such that each vertex of G has at most d neighbours coloured with the same colour as itself. We show that every planar graph is (3, 2)*-choosable and every outerplanar graph is (2, 2)*-choosable. We also propose some interesting problems about this colouring.

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