# On Domination in Graphs

@article{Gring2005OnDI, title={On Domination in Graphs}, author={Frank G{\"o}ring and Jochen Harant}, journal={Discuss. Math. Graph Theory}, year={2005}, volume={25}, pages={7-12} }

For a finite undirected graph G on n vertices two continuous optimization problems taken over the n-dimensional cube are presented and it is proved that their optimum values equal the domination number γ of G. An efficient approximation method is developed and known upper bounds on γ are slightly improved.

## 161 Citations

ON GRAPHS WHOSE NEIGHBOURHOOD CHROMATIC DOMINATION NUMBER IS TWO

- Mathematics
- 2018

In this paper we characterize the class of trees, block graphs, cactus graph, bipartite graph and different types of product graphs for which the neighbourhood chromatic domination number is equal to…

Total Dominator Colorings in Paths

- Mathematics
- 2012

Let G be a graph without isolated vertices. A total dominator coloring of a graph G is a proper coloring of the graph G with the extra property that every vertex in the graph G properly dominates a…

Random procedures for dominating sets in bipartite graphs

- MathematicsDiscuss. Math. Graph Theory
- 2010

Using multilinear functions and random procedures, new upper bounds on the domination number of a bipartite graph in terms of the cardinalities and the minimum degrees of the two colour classes are…

Dominator chromatic number of grid graph

- Mathematics
- 2020

Let G be a graph. A dominator coloring of G is a proper coloring in which every vertex of G dominates at least one color class. The dominator chromatic number of G is denoted by χd(G) and is defined…

Fair domination in the join, corona and composition of graphs

- Mathematics
- 2014

In this paper, we characterize the fair dominating sets in the join, corona and composition of graphs. We also determine the bounds or exact values of the fair domination numbers of these graphs.…

Random Procedures for Dominating Sets in Graphs

- Mathematics, Computer ScienceElectron. J. Comb.
- 2010

Several upper bounds for the domination number of a graph are derived from some random procedures for the construction of small dominating sets in graphs.

A note on the k-tuple domination number of graphs

- Mathematics
- 2021

In a graph G, a vertex dominates itself and its neighbours. A set D ⊆ V (G) is said to be a k-tuple dominating set of G if D dominates every vertex of G at least k times. The minimum cardinality…

A Survey on the Effect of Graph Operations on the Domination Number of a Graph

- Mathematics
- 2016

A dominating set of G is a set of vertices of G such that every vertex in V – D is adjacent to a vertex in D. The domination number of G, denoted by ( G ), is the minimum cardinality of a…

The Paired Domination Number of Cubic Graphs

- Mathematics
- 2020

Let G be a simple undirected graph with no isolated vertex. A paired dominating set of G is a dominating set which induces a subgraph that has a perfect matching. The paired domination number of G,…

## References

SHOWING 1-10 OF 14 REFERENCES

Some news about the independence number of a graph

- MathematicsDiscuss. Math. Graph Theory
- 2000

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…

Improved lower bounds on k-independence

- MathematicsJ. Graph Theory
- 1991

A lower bound for the maximum cardinality of a k-independent set—in terms of degree sequences—is proved which strengthens and generalizes several previously known results, including Turan's theorem.

Fundamentals of domination in graphs

- Political SciencePure and applied mathematics
- 1998

Bounds on the domination number domination, independence and irredundance efficiency, redundancy and the duals changing and unchanging domination conditions on the dominating set varieties of…

On Dominating Sets and Independent Sets of Graphs

- MathematicsCombinatorics, Probability and Computing
- 1999

The approach yields an improvement of an upper bound for the domination number found by N. Alon and J. H. Spencer, and it is proved that γk(G) = min fk(p), where the minimum is taken over the n-dimensional cube Cn.

Graph Theory

- Mathematics
- 1997

Gaph Teory Fourth Edition is standard textbook of modern graph theory which covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each chapter by one or two deeper results.

The Probabilistic Method

- Computer ScienceSODA
- 1992

A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.

Computers and Intractability: A Guide to the Theory of NP-Completeness

- Computer Science
- 1978

The experiences, understandings, and beliefs that guide the professional practices of teacher educators are explored, and the book paints a picture of a profession that offers huge rewards, alongside challenges and frustrations.

Graduate Texts in Mathematics

- Education
- 1982

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully…

Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran

- Computer Science
- 1979

Fundamentals of Domination in Graphs (Marcel

- 1998