scholarly journals A 9 k kernel for nonseparating independent set in planar graphs

2014 ◽  
Vol 516 ◽  
pp. 86-95 ◽  
Author(s):  
Łukasz Kowalik ◽  
Marcin Mucha
Keyword(s):  
Planar Graphs ◽  
Independent Set

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 $\frac{3n}{13}$.

The Strong Perfect Graph Conjecture for Planar Graphs

1973 ◽  
Vol 25 (1) ◽  
pp. 103-114 ◽  
Author(s):  
Alan Tucker
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Independent Set ◽  
Independent Sets ◽  
Clique Number ◽  
Perfect Graph ◽  
Stability Number ◽  
Partition Number ◽  
Minimum Number ◽  
The Stability

A graph G is called γ-perfect if ƛ (H) = γ(H) for every vertex-generated subgraph H of G. Here, ƛ(H) is the clique number of H (the size of the largest clique of H) and γ(H) is the chromatic number of H (the minimum number of independent sets of vertices that cover all vertices of H). A graph G is called α-perfect if α(H) = θ(H) for every vertex-generated subgraph H of G, where α (H) is the stability number of H (the size of the largest independent set of H) and θ(H) is the partition number of H (the minimum number of cliques that cover all vertices of H).

A method of graph reduction and its applications

2018 ◽  
Vol 28 (4) ◽  
pp. 249-258
Author(s):  
Dmitrii V. Sirotkin ◽  
Dmitriy S. Malyshev
Keyword(s):  
Planar Graphs ◽  
Independent Set ◽  
Simple Graph ◽  
Vertex Degree ◽  
Graph Reduction ◽  
Independent Set Problem ◽  
Np Completeness ◽  
Maximal Vertex

Abstract The independent set problem for a given simple graph is to determine the size of a maximal set of its pairwise non-adjacent vertices. We propose a new way of graph reduction leading to a new proof of the NP-completeness of the independent set problem in the class of planar graphs and to the proof of NP-completeness of this problem in the class of planar graphs having only triangular internal facets of maximal vertex degree 18.

An algorithm for finding a large independent set in planar graphs

Networks ◽  
1983 ◽  
Vol 13 (2) ◽  
pp. 247-252 ◽  
Author(s):  
Norishige Chiba ◽  
Takao Nishizeki ◽  
Nobuji Saito
Keyword(s):  
Planar Graphs ◽  
Independent Set
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