Which is harder an independent set or a vertex cover?
Independent set and Vertex Cover were proved to be equally hard, each being polynomially reducible to the other. This is due to the fact that for a given graph G, S is an independent set if and only if the set V −S (called the complement of S) is a vertex cover.
What is the matching number in independent sets?
The set of non-adjacent edges is called matching i.e independent set of edges in G such that no two edges are adjacent in the set. he parameter α 1 (G) = max { |M|: M is a matching in G } is called matching number of G i.e the maximum number of non-adjacent edges.
When is a set of vertices called an independent set?
A set of vertices I is called independent set if no two vertices in set I are adjacent to each other or in other words the set of non-adjacent vertices is called independent set. It is also called a stable set.
When is an independent set called a cover of G?
1 Independent Sets – A set of vertices I is called independent set if no two vertices in set I are adjacent to each other or in other words the 2 Vertex Covering – A set of vertices K which can cover all the edges of graph G is called a vertex cover of G i.e. 3 Matching –
When is a graph G A vertex cover?
A set of vertices C ⊆ V ( G) of a graph G is a vertex cover if and only if V ( G) ∖ C is an independent set. This is easy to see; for every endpoint of an edge, at least one vertex must be in C for C to be a vertex cover, hence not both endpoints of an edge are in V ( G) ∖ C, so V ( G) ∖ C is an independent set.
When is a set of vertices a maximal independent set?
A set of vertices is a maximal independent set if and only if it is an independent dominating set. Certain graph families have also been characterized in terms of their maximal cliques or maximal independent sets. Examples include the maximal-clique irreducible and hereditary maximal-clique irreducible graphs.
What is the complement of a maximal independent set?
The complement of a maximal independent set, that is, the set of vertices not belonging to the independent set, forms a minimal vertex cover.
How is the size of an independent set related to the number of vertices?
Therefore, the sum of the size of the largest independent set α ( G) and the size of a minimum vertex cover β ( G) is equal to the number of vertices in the graph. A vertex coloring of a graph G corresponds to a partition of its vertex set into independent subsets.
How to prove the sum of minimum edge cover and maximum?
Given a graph G, let ρ ∗ and m ∗ denote the minimum edge cover and the maximum matching of G respectively. We prove | ρ ∗ | + | m ∗ | ≤ n and | ρ ∗ | + | m ∗ | ≥ n in order below, where n is the # of vertices in G. Let S be the set of vertices that are not contained in m ∗.
How is the problem of finding an independent set solved?
The problem of finding a maximal independent set can be solved in polynomial time by a trivial greedy algorithm. All maximal independent sets can be found in time O (3 n/3) = O (1.4423 n ). exact solution.