Polynomial-time solvability of the independent set problem in a certain class of subcubic planar graphs

2017 ◽  
Vol 11 (3) ◽  
pp. 400-414
Author(s):  
D. S. Malyshev ◽  
D. V. Sirotkin
Keyword(s):  
Polynomial Time ◽  
Planar Graphs ◽  
Independent Set ◽  
Independent Set Problem

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.

scholarly journals A constructive existence theorem related to local transformations of graphs for the independent set problem

2019 ◽  
Vol 21 (2) ◽  
pp. 215-221
Author(s):  
Dmitry V. Sirotkin ◽  
Dmitry S. Malyshev
Keyword(s):  
Existence Theorem ◽  
Polynomial Time ◽  
Independence Number ◽  
Independent Set ◽  
Independent Set Problem ◽  
Power Set ◽  
Np Hardness

For a given graph, the independent set problem is to find the size of a maximum set of pairwise non-adjacent its vertices. There are numerous cases of NP-hardness and polynomial-time solvability of this problem. To determine a computational status of the independent set problem, local transformations of graphs are often used. The paper considers some class of replacements of subgraphs in graphs that change the independence number in a controllable way. Every such local transform of a graph is determined by some pattern which is a subset of the power set. It is obvious that this pattern must be gradable. The paper shows that replacing subgraph exists for any gradable pattern.

scholarly journals Independent sets in (P₆, diamond)-free graphs

2009 ◽  
Vol Vol. 11 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Raffaele Mosca
Keyword(s):  
Polynomial Time ◽  
Dominating Set ◽  
Minimum Weight ◽  
Independent Set ◽  
Independent Sets ◽  
Maximum Weight ◽  
Independent Set Problem ◽  
International Audience ◽  
Dominating Set Problem ◽  
Independent Dominating Set

Graphs and Algorithms International audience We prove that on the class of (P6,diamond)-free graphs the Maximum-Weight Independent Set problem and the Minimum-Weight Independent Dominating Set problem can be solved in polynomial time.

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