Edge dominating set
Encyclopedia
In graph theory
, 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).
for its line graph
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. 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.).
problem (and therefore finding a minimum edge dominating set is an NP-hard
problem). show that the problem is NP-complete even in the case of a bipartite graph
with maximum degree 3, and also in the case of a planar graph
with maximum degree 3.
There is a simple polynomial-time approximation algorithm
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.
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, 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 setDominating set
In graph theory, a dominating set for a graph G = is a subset D of V such that every vertex not in D is joined to at least one member of D by some edge...
for its line graph
Line graph
In graph theory, the line graph L of undirected graph G is another graph L that represents the adjacencies between edges of G...
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. 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-completeNP-complete
In computational complexity theory, the complexity class NP-complete is a class of decision problems. A decision problem L is NP-complete if it is in the set of NP problems so that any given solution to the decision problem can be verified in polynomial time, and also in the set of NP-hard...
problem (and therefore finding a minimum edge dominating set is an NP-hard
NP-hard
NP-hard , in computational complexity theory, is a class of problems that are, informally, "at least as hard as the hardest problems in NP". A problem H is NP-hard if and only if there is an NP-complete problem L that is polynomial time Turing-reducible to H...
problem). show that the problem is NP-complete even in the case of a bipartite graph
Bipartite graph
In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V; that is, U and V are independent sets...
with maximum degree 3, and also in the case of a planar graph
Planar graph
In graph theory, a planar graph is a graph that can be embedded in the 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
Approximation algorithm
In computer science and operations research, approximation algorithms are algorithms used to find approximate solutions to optimization problems. Approximation algorithms are often associated with NP-hard problems; since it is unlikely that there can ever be efficient polynomial time exact...
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.
External links
- Pierluigi Crescenzi, Viggo Kann, Magnús Halldórsson, Marek Karpinski, Gerhard Woeginger (2000), "A compendium of NP optimization problems":