# On Dominating Sets and Independent Sets of Graphs

@article{Harant1999OnDS, title={On Dominating Sets and Independent Sets of Graphs}, author={Jochen Harant and Anja Pruchnewski and Margit Voigt}, journal={Combinatorics, Probability and Computing}, year={1999}, volume={8}, pages={547 - 553} }

For a graph G on vertex set V = {1, …, n} let k = (k1, …, kn) be an integral vector such that 1 [les ] ki [les ] di for i ∈ V, where di is the degree of the vertex i in G. A k-dominating set is a set Dk ⊆ V such that every vertex i ∈ V[setmn ]Dk has at least ki neighbours in Dk. The k-domination number γk(G) of G is the cardinality of a smallest k-dominating set of G. For k1 = · · · = kn = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an…

## 56 Citations

Algorithmic Results of Independent k-Domination on Weighted Graphs

- Mathematics
- 2011

Given Given a vertex u of a connected simple graph G(V, E), let N(u) = {v | v ∈ V and (u, v) ∈ E}. We say that u dominates all vertices in N(u). Two distinct vertices u and v of G are said to be…

Improved Algorithms for k-Domination and Total k-Domination in Proper Interval Graphs

- MathematicsISCO
- 2018

This work develops faster algorithms for k-domination and total k-Domination in the class of proper interval graphs that run in time \(\mathcal {O}(|V(G)|^{3k})\) for each fixed \(k\ge 1\) and are also applicable to the weighted case.

Restricted domination parameters in graphs

- MathematicsJ. Comb. Optim.
- 2007

For a property π of subsets of V(G), with associated parameter f_π, the k-restricted π-number rk(G,f_π) is the smallest integer r such that given any subset K of k vertices of G, there exists a π set containing K of (at most) cardinality r.

On Double Domination in Graphs

- MathematicsDiscuss. Math. Graph Theory
- 2005

If G has order n with minimum degree ‐ and average degree d, then ∞£2(G) • ((ln(1 + d) + ln‐ + 1)=‐)n, where the minimum is taken over the n-dimensional cube C n.

(Total) Vector domination for graphs with bounded branchwidth

- MathematicsDiscret. Appl. Math.
- 2016

On 2-Step and Hop Dominating Sets in Graphs

- MathematicsGraphs Comb.
- 2017

It is proved that almost all graphs have a hop dominating set of cardinality at most the total domination number if p is constant, and that the decision problems for the 2-step dominating set andHop dominating set problems are NP-complete for planar bipartite graphs and planar chordal graphs.

New Algorithms for Weighted k-Domination and Total k-Domination Problems in Proper Interval Graphs

- MathematicsTheor. Comput. Sci.
- 2019

(Total) Vector Domination for Graphs with Bounded Branchwidth

- Mathematics
- 2014

Given a graphG = (V,E) of ordern and ann-dimensional non-negative vector d = (d(1),d(2), . . . ,d(n)), called demand vector, the vector domination (resp., total vector domination) is the problem of…

Vector connectivity in graphs

- Mathematics, Computer ScienceTAMC
- 2013

It is shown that VECTOR CONNECTIVITY can actually be solved in polynomial time on split graphs, in addition to cographs and trees, and that the problem can be approximated in poynomial time within a factor of lnn+2 on all n‐vertex graphs.

## References

SHOWING 1-10 OF 24 REFERENCES

On the integrity of distance domination in graphs

- MathematicsAustralas. J Comb.
- 1994

It is shown that the problem of computing 'Yn,k is in the NP-complete class, even when restricted to bipartite graphs and chordal graphs, and it is proved that in every graph there exist some subsets of vertices that are both (n, k )-independent and ( n, k)-dominating.

A new domination conception

- MathematicsJ. Graph Theory
- 1993

Est estimations of the new domination number μj,k(G) are given, and with the help of these estimations some new and some known upper bounds for the j-domination number are proved.

On a conjecture of Fink and Jacobson concerning k-domination and k-dependence

- MathematicsJ. Comb. Theory, Ser. B
- 1985

An upper bound for the k-domination number of a graph

- MathematicsJ. Graph Theory
- 1985

The k-domination number of a graph G, γk(G), is the least cardinality of a set U of verticies such that any other vertex is adjacent to at least k vertices of U. We prove that if each vertex has…

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.

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

- Computer Science
- 1978

Horn formulae play a prominent role in artificial intelligence and logic programming. In this paper we investigate the problem of optimal compression of propositional Horn production rule knowledge…

Dominating Sets and Independent Sets