scholarly journals Disjoint Paths in a Planar Graph—A General Theorem

1992 ◽  
Vol 5 (1) ◽  
pp. 112-116 ◽  
Author(s):  
Guoli Ding ◽  
A. Schrijver ◽  
P. D. Seymour
Keyword(s):  
Planar Graph ◽  
General Theorem ◽  
Disjoint Paths

scholarly journals 3-Connected Cores In Random Planar Graphs

2011 ◽  
Vol 20 (3) ◽  
pp. 381-412 ◽  
Author(s):  
NIKOLAOS FOUNTOULAKIS ◽  
KONSTANTINOS PANAGIOTOU
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
General Theorem ◽  
Building Blocks ◽  
Discrete Mathematics ◽  
List Type ◽  
Type Number ◽  
Side Constraints ◽  
Evaluating Methods ◽  
Sharp Concentration

The study of the structural properties of large random planar graphs has become in recent years a field of intense research in computer science and discrete mathematics. Nowadays, a random planar graph is an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints.In this paper we focus on the structure of random 2-connected planar graphs regarding the sizes of their 3-connected building blocks, which we callcores. In fact, we prove a general theorem regarding random biconnected graphs from various classes. IfBnis a graph drawn uniformly at random from a suitable classof labelled biconnected graphs, then we show that with probability 1 −o(1) asn→ ∞,Bnbelongs to exactly one of the following categories:(i)either there is a uniquegiantcore inBn, that is, there is a 0 <c=c() < 1 such that the largest core contains ~cnvertices, and every other core contains at mostnαvertices, where 0 < α = α() < 1;(ii)or all cores ofBncontainO(logn) vertices.Moreover, we find the critical condition that determines the category to whichBnbelongs, and also provide sharp concentration results for the counts of cores of all sizes between 1 andn. As a corollary, we obtain that a random biconnected planar graph belongs to category (i), where in particularc= 0.765. . . and α = 2/3.

scholarly journals On the Maximum Degree of Path-Pairable Planar Graphs

10.37236/7291 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
António Girão ◽  
Gábor Mészáros ◽  
Kamil Popielarz ◽  
Richard Snyder
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Disjoint Paths ◽  
Edge Disjoint ◽  
Finite Genus

A graph is path-pairable if for any pairing of its vertices there exist edge-disjoint paths joining the vertices in each pair. We investigate the behaviour of the maximum degree in path-pairable planar graphs. We show that any $n$-vertex path-pairable planar graph must contain a vertex of degree linear in $n$. Our result generalizes to graphs embeddable on a surface of finite genus.  

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 A tight Erdős-Pósa function for planar minors

2019 ◽  
Author(s):  
Wouter Cames van Batenburg ◽  
Tony Huyn ◽  
Gwenaël Joret ◽  
Jean-Florent Raymond
Keyword(s):  
Planar Graph ◽  
General Theorem ◽  
Large Body ◽  
Minimum Size ◽  
Property A ◽  
Disjoint Cycles ◽  
The Family ◽  
Odd Cycle ◽  
A Minor ◽  
Odd Cycles

Let F be a family of graphs. Then for every graph G the maximum number of disjoint subgraphs of G, each isomorphic to a member of F, is at most the minimum size of a set of vertices that intersects every subgraph of G isomorphic to a member of F. We say that F packs if equality holds for every graph G. Only very few families pack. As the next best weakening we say that F has the Erdős-Pósa property if there exists a function f such that for every graph G and integer k>0 the graph G has either k disjoint subgraphs each isomorphic to a member of F or a set of at most f(k) vertices that intersects every subgraph of G isomorphic to a member of F. The name is motivated by a classical 1965 result of Erdős and Pósa stating that for every graph G and integer k>0 the graph G has either k disjoint cycles or a set of O(klogk) vertices that intersects every cycle. Thus the family of all cycles has the Erdős-Pósa property with f(k)=O(klogk). In contrast, the family of odd cycles fails to have the Erdős-Pósa property. For every integer ℓ, a sufficiently large Escher Wall has an embedding in the projective plane such that every face is even and every homotopically non-trivial closed curve intersects the graph at least ℓ times. In particular, it contains no set of ℓ vertices such that each odd cycle contains at least one them, yet it has no two disjoint odd cycles. By now there is a large body of literature proving that various families F have the Erdős-Pósa property. A very general theorem of Robertson and Seymour says that for every planar graph H the family F(H) of all graphs with a minor isomorphic to H has the Erdős-Pósa property. (When H is non-planar, F(H) does not have the Erdős-Pósa property.) The present paper proves that for every planar graph H the family F(H) has the Erdős-Pósa property with f(k)=O(klogk), which is asymptotically best possible for every graph H with at least one cycle.

scholarly journals Finding k Disjoint Paths in a Directed Planar Graph

1994 ◽  
Vol 23 (4) ◽  
pp. 780-788 ◽  
Author(s):  
Alexander Schrijver
Keyword(s):  
Planar Graph ◽  
Disjoint Paths

scholarly journals Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ2,1,2

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 708
Author(s):  
Donghan Zhang
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Total Coloring ◽  
Disjoint Paths ◽  
Combinatorial Nullstellensatz ◽  
Proper Total Coloring

A theta graph Θ2,1,2 is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring ϕ of G is a proper total coloring of G such that ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each edge uv∈E(G), where EG(u) denotes the set of edges incident with a vertex u. In 2015, Pilśniak and Woźniak introduced this coloring and conjectured that every graph with maximum degree Δ admits an NSD total (Δ+3)-coloring. In this paper, we show that the listing version of this conjecture holds for any IC-planar graph with maximum degree Δ≥9 but without theta graphs Θ2,1,2 by applying the Combinatorial Nullstellensatz, which improves the result of Song et al.

scholarly journals Lattice Structures from Planar Graphs

10.37236/1768 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Stefan Felsner
Keyword(s):  
Distributive Lattice ◽  
Planar Graph ◽  
Planar Graphs ◽  
Spanning Trees ◽  
Lattice Structure ◽  
General Theorem ◽  
Lattice Structures ◽  
Schnyder Woods

The set of all orientations of a planar graph with prescribed outdegrees carries the structure of a distributive lattice. This general theorem is proven in the first part of the paper. In the second part the theorem is applied to show that interesting combinatorial sets related to a planar graph have lattice structure: Eulerian orientations, spanning trees and Schnyder woods. For the Schnyder wood application some additional theory has to be developed. In particular it is shown that a Schnyder wood for a planar graph induces a Schnyder wood for the dual.

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