scholarly journals 3‐Degenerate induced subgraph of a planar graph

2021 ◽  
Author(s):  
Yangyan Gu ◽  
H. A. Kierstead ◽  
Sang‐il Oum ◽  
Hao Qi ◽  
Xuding Zhu
Keyword(s):  
Planar Graph ◽  
Induced Subgraph

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.

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.

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.

Computational Micrograph Registration with Sieve Processes

1996 ◽  
Vol 54 ◽  
pp. 440-441
Author(s):  
P.J. Phillips ◽  
J. Huang ◽  
S. M. Dunn
Keyword(s):  
Planar Graph ◽  
Efficient Algorithm ◽  
Spatial Organization ◽  
Spatial Arrangement ◽  
Stereo Image ◽  
Image Model ◽  
Geometric Invariants ◽  
Point Set

In this paper we present an efficient algorithm for automatically finding the correspondence between pairs of stereo micrographs, the key step in forming a stereo image. The computation burden in this problem is solving for the optimal mapping and transformation between the two micrographs. In this paper, we present a sieve algorithm for efficiently estimating the transformation and correspondence.In a sieve algorithm, a sequence of stages gradually reduce the number of transformations and correspondences that need to be examined, i.e., the analogy of sieving through the set of mappings with gradually finer meshes until the answer is found. The set of sieves is derived from an image model, here a planar graph that encodes the spatial organization of the features. In the sieve algorithm, the graph represents the spatial arrangement of objects in the image. The algorithm for finding the correspondence restricts its attention to the graph, with the correspondence being found by a combination of graph matchings, point set matching and geometric invariants.

scholarly journals A Thomassen-type method for planar graph recoloring

2021 ◽  
Vol 95 ◽  
pp. 103319
Author(s):  
Zdeněk Dvořák ◽  
Carl Feghali
Keyword(s):  
Planar Graph ◽  
Type Method

scholarly journals Toughness, Forbidden Subgraphs and Pancyclicity

2021 ◽  
Vol 37 (3) ◽  
pp. 839-866
Author(s):  
Wei Zheng ◽  
Hajo Broersma ◽  
Ligong Wang
Keyword(s):  
Recent Article ◽  
Forbidden Subgraph ◽  
Forbidden Subgraphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Complete Answer ◽  
Open Case

AbstractMotivated by several conjectures due to Nikoghosyan, in a recent article due to Li et al., the aim was to characterize all possible graphs H such that every 1-tough H-free graph is hamiltonian. The almost complete answer was given there by the conclusion that every proper induced subgraph H of $$K_1\cup P_4$$ K 1 ∪ P 4 can act as a forbidden subgraph to ensure that every 1-tough H-free graph is hamiltonian, and that there is no other forbidden subgraph with this property, except possibly for the graph $$K_1\cup P_4$$ K 1 ∪ P 4 itself. The hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs, as conjectured by Nikoghosyan, was left there as an open case. In this paper, we consider the stronger property of pancyclicity under the same condition. We find that the results are completely analogous to the hamiltonian case: every graph H such that any 1-tough H-free graph is hamiltonian also ensures that every 1-tough H-free graph is pancyclic, except for a few specific classes of graphs. Moreover, there is no other forbidden subgraph having this property. With respect to the open case for hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs we give infinite families of graphs that are not pancyclic.

Plane embeddings of planar graph metrics

2006 ◽  
Author(s):  
MohammadHossein Bateni ◽  
MohammadTaghi Hajiaghayi ◽  
Erik D. Demaine ◽  
Mohammad Moharrami
Keyword(s):  
Planar Graph ◽  
Graph Metrics ◽  
Plane Embeddings

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.

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