# Some news about the independence number of a graph

@article{Harant2000SomeNA, title={Some news about the independence number of a graph}, author={Jochen Harant}, journal={Discuss. Math. Graph Theory}, year={2000}, volume={20}, pages={71-79} }

For a finite undirected graph G on n vertices some continuous optimization problems taken over the n-dimensional cube are presented and it is proved that their optimum values equal the independence number of G.

## 21 Citations

On Domination in Graphs

- MathematicsDiscuss. Math. Graph Theory
- 2005

It is proved that their optimum values equal the domination number γ of G, and an efficient approximation method is developed and known upper bounds on γ are slightly improved.

On a Polynomial Fractional Formulation for Independence Number of a Graph

- MathematicsJ. Glob. Optim.
- 2006

Classical Karush-Kuhn-Tucker conditions and simple combinatorial arguments are found sufficient to deduce several interesting properties of the local and global maxima of a continuous global optimization formulation for finding the independence number of a graph.

Clique , independent set , and graph coloring

- Mathematics
- 2010

This article introduces the closely related maximum clique, maximum independent set, graph coloring, and minimum clique partitioning problems. The survey includes some of the most important results…

A Heuristic for the Maximum Independent Set Problem Based on Optimization of a Quadratic Over a Sphere

- MathematicsJ. Comb. Optim.
- 2002

A heuristic for the maximum independent set problem is proposed which utilizes classical results for the problem of optimization of a quadratic function over a sphere and its efficiency is confirmed by results of numerical experiments on DIMACS benchmarks.

Maximum Clique, Maximum Independent Set, and Graph Coloring Problems

- Mathematics
- 2011

This article introduces the closely related maximum clique, maximum independent set, graph coloring, and minimum clique partitioning problems. The survey includes some of the most important results…

Finding independent sets in a graph using continuous multivariable polynomial formulations

- MathematicsJ. Glob. Optim.
- 2001

This work proposes two polynomial-time algorithms, based on two continuous formulations of the maximum independent set problem on a graph G=(V,E), for finding maximal independent sets with cardinality greater than or equal to F( x0) and H(x0), respectively.

Finding the Maximal Independent Sets of a Graph Including the Maximum Using a Multivariable Continuous Polynomial Objective Optimization Formulation

- Computer Science, MathematicsSAI
- 2020

This work proposes a multivariable continuous polynomial optimization formulation to find arbitrary maximal independent sets of any size for any graph and believes that this algorithm is efficient for sparse graphs, for which there exist fast algorithms to list their maximal cliques.

A linear complementarity based characterization of the weighted independence number and the independent domination number in graphs

- MathematicsDiscret. Appl. Math.
- 2018

Constructing test functions for global optimization using continuous formulations of graph problems

- MathematicsOptim. Methods Softw.
- 2005

A method for constructing test functions for global optimization which utilizes continuous formulations of combinatorial optimization problems is suggested, and a number of sample test functions based on these formulations are proposed.

On characterization of maximal independent sets via quadratic optimization

- MathematicsJ. Heuristics
- 2013

Theoretical results characterizing binary local maxima in terms of certain induced subgraphs of the given graph are developed and these results are used to develop an efficient local search algorithm that provides considerable speed-up over a typical local search algorithms for the binary Hamming distance-2 neighborhood.

## References

SHOWING 1-10 OF 24 REFERENCES

A lower bound on the independence number of a graph

- Mathematics, Computer ScienceDiscret. Math.
- 1998

On the Independence Number of a Graph in Terms of N and M

- Mathematics

For the independence number (G) of a connected graph G on n vertices with m edges the inequality (G) 1 2 (2m + n + 1) ? p (2m + n + 1) 2 ? 4n 2 ] is proved and its algorithmic realization is…

A Probabilistic lower bound on the independence number of graphs

- Mathematics, Computer ScienceDiscret. Math.
- 1994

Independence, clique size and maximum degree

- MathematicsComb.
- 1984

It was shown before that ifG is a graph of maximum degreep containing no cliques of the sizeq then the independence ratio is greater than or equal to 2 / (p +q). We shall discuss here some extreme…

The independence number of graphs in terms of degrees

- Mathematics, Computer ScienceDiscret. Math.
- 1993

Maxima for Graphs and a New Proof of a Theorem of Turán

- MathematicsCanadian Journal of Mathematics
- 1965

Maximum of a square-free quadratic form on a simplex. The following question was suggested by a problem of J. E. MacDonald Jr. (1): Given a graph G with vertices 1, 2, . . . , n. Let S be the simplex…

A note on the independence number of triangle-free graphs, II

- MathematicsJ. Comb. Theory, Ser. B
- 1991

Approximating Maximum Independent Sets by Excluding Subgraphs

- MathematicsBIT
- 1990

An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known to n/(log n)2, and this can be combined into a surprisingly strong simultaneous performance guarantee for the clique and coloring problems.