graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, an edge dominating set for a graph ''G'' = (''V'', ''E'') is a subset ''D'' ⊆ ''E'' such that every edge not in ''D'' is adjacent to at least one edge in ''D''. An edge dominating set is also known as a ''line dominating set''. Figures (a)–(d) are examples of edge dominating sets (thick red lines). A minimum edge dominating set is a smallest edge dominating set. Figures (a) and (b) are examples of minimum edge dominating sets (it can be checked that there is no edge dominating set of size 2 for this graph).

Properties

An edge dominating set for ''G'' is a

dominating set In graph theory, a dominating set for a Graph (discrete mathematics), graph is a subset of its vertices, such that any vertex of is in , or has a neighbor in . The domination number is the number of vertices in a smallest dominating set for ...

for its

line graph In the mathematics, mathematical discipline of graph theory, the line graph of an undirected graph is another graph that represents the adjacencies between edge (graph theory), edges of . is constructed in the following way: for each edge i ...

''L''(''G'') and vice versa. Any maximal matching is always an edge dominating set. Figures (b) and (d) are examples of maximal matchings. Furthermore, the size of a minimum edge dominating set equals the size of a

minimum maximal matching In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge ...

. A minimum maximal matching is a minimum edge dominating set; Figure (b) is an example of a minimum maximal matching. A minimum edge dominating set is not necessarily a minimum maximal matching, as illustrated in Figure (a); however, given a minimum edge dominating set ''D'', it is easy to find a minimum maximal matching with , ''D'', edges (see, e.g., ).

Algorithms and computational complexity

Determining whether there is an edge dominating set of a given size for a given graph is an

NP-complete In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any ...

problem (and therefore finding a minimum edge dominating set is an

NP-hard In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assumi ...

problem). show that the problem is NP-complete even in the case of a

bipartite graph In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theo ...

with maximum degree 3, and also in the case of a

planar graph In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. ...

with maximum degree 3. There is a simple polynomial-time

approximation algorithm In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned sol ...

with approximation factor 2: find any maximal matching. A maximal matching is an edge dominating set; furthermore, a maximal matching ''M'' can be at worst 2 times as large as a smallest maximal matching, and a smallest maximal matching has the same size as the smallest edge dominating set. Also the edge-weighted version of the problem can be approximated within factor 2, but the algorithm is considerably more complicated (; ). show that finding a better than (7/6)-approximation is NP-hard. Schmied & Viehmann proved that the Problem is UGC-hard to approximate to within any constant better than 3/2.

References

*. ::Minimum edge dominating set (optimisation version) is the problem GT3 in Appendix B (page 370). ::Minimum maximal matching (optimisation version) is the problem GT10 in Appendix B (page 374). * . *. ::Edge dominating set (decision version) is discussed under the dominating set problem, which is the problem GT2 in Appendix A1.1. ::Minimum maximal matching (decision version) is the problem GT10 in Appendix A1.1. * . * . * {{citation , last1=Parekh , first1=Ojas , contribution=Edge dominating and hypomatchable sets , title=Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms , year=2002 , pages=287–291 , contribution-url=http://dl.acm.org/citation.cfm?id=545381.545419 . *Richard Schmied & Claus Viehmann (2012), "Approximating edge dominating set in dense graphs", Theor. Comput. Sci. 414(1), pp. 92-99.

External links

* Pierluigi Crescenzi, Viggo Kann, Magnús Halldórsson, Marek Karpinski, Gerhard Woeginger (2000)
"A compendium of NP optimization problems"
:

:

NP-complete problems Computational problems in graph theory