Dominating set

In graph theory, a dominating set for a graph G = (V, E) is a subset D of V such that every vertex not in D is adjacent to at least one member of D. The domination number γ(G) is the number of vertices in a smallest dominating set for G. In graph theory, a dominating set for a graph G = (V, E) is a subset D of V such that every vertex not in D is adjacent to at least one member of D. The domination number γ(G) is the number of vertices in a smallest dominating set for G. The dominating set problem concerns testing whether γ(G) ≤ K for a given graph G and input K; it is a classical NP-complete decision problem in computational complexity theory. Therefore it is believed that there may be no efficient algorithm that finds a smallest dominating set for all graphs, although there are efficient approximation algorithms, as well as both efficient and exact algorithms for certain graph classes. Figures (a)–(c) on the right show three examples of dominating sets for a graph. In each example, each white vertex is adjacent to at least one red vertex, and it is said that the white vertex is dominated by the red vertex. The domination number of this graph is 2: the examples (b) and (c) show that there is a dominating set with 2 vertices, and it can be checked that there is no dominating set with only 1 vertex for this graph. The domination problem was studied from the 1950s onwards, but the rate of research on domination significantly increased in the mid-1970s. In 1972, Richard Karp proved the set cover problem to be NP-complete. This had immediate implications for the dominating set problem, as there are straightforward vertex to set and edge to non-disjoint-intersection bijections between the two problems. This proved the dominating set problem to be NP-complete as well. Dominating sets are of practical interest in several areas. In wireless networking, dominating sets are used to find efficient routes within ad-hoc mobile networks. They have also been used in document summarization, and in designing secure systems for electrical grids. Dominating sets are closely related to independent sets: an independent set is also a dominating set if and only if it is a maximal independent set, so any maximal independent set in a graph is necessarily also a minimal dominating set. Thus, the smallest maximal independent set is greater or equal in size than the smallest independent dominating set. The independent domination number i(G) of a graph G is the size of the smallest independent dominating set (or, equivalently, the size of the smallest maximal independent set). The minimum dominating set in a graph will not necessarily be independent, but the size of a minimum dominating set is always less than or equal to the size of a minimum maximal independent set, that is, γ(G) ≤ i(G). There are graph families in which a minimum maximal independent set is a minimum dominating set. For example, γ(G) = i(G) if G is a claw-free graph. A graph G is called a domination-perfect graph if γ(H) = i(H) in every induced subgraph H of G. Since an induced subgraph of a claw-free graph is claw-free, it follows that every claw-free graphs is also domination-perfect.