scholarly journals Maximum 4-Degenerate Subgraph of a Planar Graph

10.37236/4265 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Robert Lukot'ka ◽  
Ján Mazák ◽  
Xuding Zhu
Keyword(s):  
Planar Graph ◽  
Local Variation ◽  
Average Degree ◽  
Induced Subgraph ◽  
Empty Graph ◽  
Connected Planar Graph

A graph $G$ is $k$-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree $k$ or less. We prove that every connected planar graph with average degree $d \ge  2$ has a $4$-degenerate induced subgraph containing at least $ (38 - d)/36$ of its vertices. This shows that every planar graph of order $n$ has a $4$-degenerate induced subgraph of order more than $8/9 \cdot n$. We also consider a local variation of this problem and show that in every planar graph with at least 7 vertices, deleting a suitable vertex allows us to subsequently remove at least 6 more vertices of degree four or less.

The random division of faces in a planar graph

1996 ◽  
Vol 28 (2) ◽  
pp. 331-331
Author(s):  
Richard Cowan ◽  
Simone Chen
Keyword(s):  
Planar Graph ◽  
Limiting Distribution ◽  
Face Type ◽  
Connected Planar Graph ◽  
Random Division

Consider a connected planar graph. A bounded face is said to be of type k, or is called a k-face, if the boundary of that face contains k edges. Under various natural rules for randomly dividing bounded faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

scholarly journals Rooted $K_4$-Minors

10.37236/3476 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Ruy Fabila-Monroy ◽  
David R. Wood
Keyword(s):  
Planar Graph ◽  
Pairwise Disjoint ◽  
Single Face ◽  
Special Cases ◽  
Connected Subgraphs ◽  
Connected Planar Graph

Let $a,b,c,d$ be four vertices in a graph $G$. A $K_4$ minor rooted at $a,b,c,d$ consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of $G$, respectively containing $a,b,c,d$. We characterise precisely when $G$ contains a $K_4$-minor rooted at $a,b,c,d$ by describing six classes of obstructions, which are the edge-maximal graphs containing no $K_4$-minor rooted at $a,b,c,d$. The following two special cases illustrate the full characterisation: (1) A 4-connected non-planar graph contains a $K_4$-minor rooted at $a,b,c,d$ for every choice of $a,b,c,d$. (2) A 3-connected planar graph contains a $K_4$-minor rooted at $a,b,c,d$ if and only if $a,b,c,d$ are not on a single face.

The random division of faces in a planar graph

1996 ◽  
Vol 28 (02) ◽  
pp. 331
Author(s):  
Richard Cowan ◽  
Simone Chen
Keyword(s):  
Planar Graph ◽  
Limiting Distribution ◽  
Face Type ◽  
Connected Planar Graph ◽  
Random Division

Consider a connected planar graph. A bounded face is said to be of type k, or is called a k-face, if the boundary of that face contains k edges. Under various natural rules for randomly dividing bounded faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

scholarly journals Adjacency Labelling for Planar Graphs (and Beyond)

2021 ◽  
Vol 68 (6) ◽  
pp. 1-33
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Cyril Gavoille ◽  
Gwenaël Joret ◽  
Piotr Micek ◽  
...  
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Order Term ◽  
Optimal Result ◽  
Induced Subgraph ◽  
Bounded Degree ◽  
Graph Classes ◽  
Bounded Degree Graphs ◽  
Minor Free Graphs ◽  
Labelling Scheme

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an n -vertex planar graph G is assigned a (1 + o(1)) log 2 n -bit label and the labels of two vertices u and v are sufficient to determine if uv is an edge of G . This is optimal up to the lower order term and is the first such asymptotically optimal result. An alternative, but equivalent, interpretation of this result is that, for every positive integer n , there exists a graph U n with n 1+o(1) vertices such that every n -vertex planar graph is an induced subgraph of U n . These results generalize to a number of other graph classes, including bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and k -planar graphs.

On optimization of of network traffic according to the communication matrix

2021 ◽  
Vol 6 ◽  
Author(s):  
Yu.M. Bogdanov ◽  
◽  
S.A. Selivanov ◽  
A.V. Sinitsyn ◽  
A.A. Sinitsyn ◽  
...  
Keyword(s):  
Planar Graph ◽  
Network Traffic ◽  
Virtual Network ◽  
Software Defined Network ◽  
Network Function ◽  
Multiservice Network ◽  
Network Equipment ◽  
Switching Matrix ◽  
Connected Planar Graph ◽  
Virtual Network Function

The article is devoted to the search for solutions to optimize multiservice network traffic taking according the switching matrix by increasing the capabilities of network equipment with restrictions on their number with the ability to implement the algorithm as a Virtual Network Function for a Software-Defined Network. It is assumed that the networks are represented by an undirected complete connected planar graph.

scholarly journals On the Diameter of Random Planar Graphs

2014 ◽  
Vol 24 (1) ◽  
pp. 145-178 ◽  
Author(s):  
GUILLAUME CHAPUY ◽  
ÉRIC FUSY ◽  
OMER GIMÉNEZ ◽  
MARC NOY
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Formula Group ◽  
Image Position ◽  
Connected Planar Graph

We show that the diameter diam(Gn) of a random labelled connected planar graph withnvertices is equal ton1/4+o(1), in probability. More precisely, there exists a constantc> 0 such that$$ P(\D(G_n)\in(n^{1/4-\e},n^{1/4+\e}))\geq 1-\exp(-n^{c\e}) $$for ε small enough andn ≥ n0(ε). We prove similar statements for 2-connected and 3-connected planar graphs and maps.

scholarly journals Restricted Frame Graphs and a Conjecture of Scott

10.37236/4424 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Jérémie Chalopin ◽  
Louis Esperet ◽  
Zhentao Li ◽  
Patrice Ossona de Mendez
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Clique Number ◽  
Induced Subgraph ◽  
Intersection Graphs

Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is replaced by any graph $H$. Pawlik et al. recently constructed a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erdős). This shows that Scott's conjecture is false whenever $H$ is obtained from a non-planar graph by subdividing every edge at least once.It remains interesting to decide which graphs $H$ satisfy Scott's conjecture and which do not. In this paper, we study the construction of Pawlik et al. in more details to extract more counterexamples to Scott's conjecture. For example, we show that Scott's conjecture is false for any graph obtained from $K_4$ by subdividing every edge at least once.  We also prove that if $G$ is a 2-connected multigraph with no vertex contained in every cycle of $G$, then any graph obtained from $G$ by subdividing every edge at least twice is a counterexample to Scott's conjecture.

Directed Projection Graph of N-Dimensional Hypercube and Subhypercube Decomposition of Balanced Linearly Separable Boolean Functions

2021 ◽  
Vol 31 (09) ◽  
pp. 2150138
Author(s):  
Wei Jin ◽  
Fangyue Chen ◽  
Qinbin He
Keyword(s):  
Neural Networks ◽  
Artificial Neural Networks ◽  
Boolean Function ◽  
Planar Graph ◽  
Boolean Functions ◽  
Conformal Transformation ◽  
Two Dimensional ◽  
Induced Subgraph ◽  
Projection Graph ◽  
Enumeration Scheme

A directed projection graph of the [Formula: see text]-dimensional hypercube on the two-dimensional plane is successfully created. Any [Formula: see text]-variable Boolean function can be easily transformed to an induced subgraph of the projection. Therefore, the discussions on [Formula: see text]-variable Boolean functions only need to focus on a two-dimensional planar graph. Some mathematical theories on the projection graph and the induced subgraph are established, and some properties and characteristics of a balanced linearly separable Boolean function (BLSBF) are uncovered. In particular, the sub-hypercube decompositions of BLSBF is easily represented on the projection, and meanwhile, the enumeration scheme for counting the number of [Formula: see text]-variable BLSBFs is developed by using equivalence classification and conformal transformation. With the aid of the directed projection grap constructed in this paper, one can further study many difficult problems in some fields such as Boolean functions and artificial neural networks.

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