What is a dominating set in graph theory?
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. 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.
Who introduced domination in graphs?
Oystein Ore [39] introduced the terms “dominating set” and “domination number” in his book on graph theory which was published in 1962.
What are dominant numbers?
The (lower) domination number of a graph is the minimum size of a dominating set of vertices in , i.e., the size of a minimum dominating set. This is equivalent to the smallest size of a minimal dominating set since every minimum dominating set is also minimal.
What is the study of graph theory?
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines).
What is total dominating set?
A set S of vertices in a graph G(V,E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. Respectively the total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G.
How do I find my domination number?
A vertex set of a graph is a dominating set if each vertex of either belongs to or is adjacent to a vertex in . The domination number of is the minimum cardinality of as varies over all dominating sets of . It is known that γ ( G ) ≥ 1 3 ( d i a m ( G ) + 1 ) , where d i a m ( G ) denotes the diameter of .
What is minimum connected dominating set?
A Minimum Connected Dominating Set is a minimum set of connected nodes such that every other node in the network is one hop connected with a node in this set. In general, the problemis proved to be NP-hard. In this paper we find a Minimum Connected Dominating Set for certain Circulant Networks.
What is independent domination?
1. Introduction. An independent dominating set in a graph is a set that is both dominating and independent. A dominating set of a graph is a set of vertices of such that every vertex not in is adjacent to a vertex in . The domination number of , denoted by , is the minimum size of a dominating set.
How graph can help in managing the information?
Graphs are a common method to visually illustrate relationships in the data. The purpose of a graph is to present data that are too numerous or complicated to be described adequately in the text and in less space. Do not, however, use graphs for small amounts of data that could be conveyed succinctly in a sentence.
Is dominating set NP-hard?
Dominating Set is NP-Hard Every instance of the Vertex Cover problem consists of a graph G = (V, E) and an integer k consisting of the subset of vertices as the input can be converted to a Dominating Set problem consisting of graph G’ = (V’, E’).
How do you find the maximal independent set of a graph?
For a graph G = (V,E), an independent set is a set S ⊂ V which contains no edges of G, i.e., for all (u, v) ∈ E either u ∈ S and/or v ∈ S. The independent set S is a maximal independent set if for all v ∈ V , either v ∈ S or N(v) ∩ S = ∅ where N(v) denotes the neighbors of v.