# Efficient Dominating and Edge Dominating Sets for Graphs and Hypergraphs

```@inproceedings{Brandstdt2016EfficientDA,
title={Efficient Dominating and Edge Dominating Sets for Graphs and Hypergraphs},
author={Andreas Brandst{\"a}dt and Ragnar Nevries},
booktitle={Encyclopedia of Algorithms},
year={2016}
}```
• Published in Encyclopedia of Algorithms 4 July 2012
• Mathematics
Let G = (V,E) be a graph. A vertex dominates itself and all its neighbors, i.e., every vertex v ∈ V dominates its closed neighborhood N[v]. A vertex set D in G is an efficient dominating (e.d.) set for G if for every vertex v ∈ V, there is exactly one d ∈ D dominating v. An edge set M ⊆ E is an efficient edge dominating (e.e.d.) set for G if it is an efficient dominating set in the line graph L(G) of G. The ED problem (EED problem, respectively) asks for the existence of an e.d. set (e.e.d. set…
22 Citations
Exact algorithms for dominating induced matching based on graph partition
• Mathematics, Computer Science
Discret. Appl. Math.
• 2015
On resolving efficient domination number of comb product of special graphs
• Mathematics
• 2021
Let G be a connected, finite, and undirected graph. A vertex set D in G is an efficient dominating set of G if D is an independent set and for each point υ ∈ V(G)-D is adjacent to precisely one
On Efficient Domination for Some Classes of H-Free Chordal Graphs
• Mathematics
Electron. Notes Discret. Math.
• 2017
Efficient Domination for Some Subclasses of P_6 -free Graphs in Polynomial Time
• Mathematics
WG
• 2015
It is shown that squares of P_6-free chordal graphs that have an e.d. are hole-free and some properties concerning odd antiholes show that ED/WeightedED is solvable in polynomial time for P_ 6, house, hole, domino-free graphs, which implies that the time bound for P-6, house- free graphs is better than that for the other graphs.
Dominating Induced Matchings for P8-free Graphs in Polynomial Time
• Mathematics
ArXiv
• 2015
It is shown in this paper that the weighted DIM problem is solvable in polynomial time for \$P_8\$-free graphs.
Perfect edge domination: hard and solvable cases
• Mathematics
Ann. Oper. Res.
• 2018
Two NP-completeness proofs are described and a complexity dichotomy theorem is proved for the perfect edge domination problem, and a linear time algorithm for finding a minimum weight perfect edge dominating set of a \$\$P_5\$\$P5-free graph is described.
Finding Efficient Domination for \$(S_{1,2,5},S_{3,3,3}\$-Free Chordal Bipartite Graphs in Polynomial Time
Here it is shown that ED can be solved in polynomial time for (S1,2,5, S2,3,3)-free chordal bipartite graphs.
Finding Dominating Induced Matchings in \$\$P_8\$\$P8-Free Graphs in Polynomial Time
• Mathematics
Algorithmica
• 2016
The weighted DIM problem is solvable in polynomial time for P_8-free graphs, which is NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three.

## References

SHOWING 1-10 OF 54 REFERENCES
Perfect edge domination and efficient edge domination in graphs
• Mathematics, Computer Science
Discret. Appl. Math.
• 2002
The Weighted Perfect Domination Problem and Its Variants
• Mathematics, Computer Science
Discret. Appl. Math.
• 1996
Dominating Induced Matchings for P7-Free Graphs in Linear Time
• Mathematics
Algorithmica
• 2012
The weighted DIM problem is solvable in linear time for P7-free graphs in a robust way and is-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three.
Efficient edge domination in regular graphs
• Mathematics
Discret. Appl. Math.
• 2008
Efficient Edge Domination Problems in Graphs
• Mathematics, Computer Science
Inf. Process. Lett.
• 1993