# What is maximal independent set in graph theory?

## What is maximal independent set in graph theory?

In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independent set that may join it because it is maximal with respect to the independent set property.

## How do you find the maximal independent set of a graph?

Maximum Independent Vertex Set In a complete graph, each vertex is adjacent to its remaining (n − 1) vertices. Therefore, a maximum independent set of Kn contains only one vertex. If ‘S’ is an independent vertex set of ‘G’, then (V – S) is a vertex cover of G.

What is the difference between a maximum independent set and a maximal independent set in a graph?

A maximal independent set of a graph G is an independent set which is not contained properly in any other independent set of G. An independent set is called maximum if it is of largest cardinality.

What does maximal mean in graph theory?

largest possible
Maximal means that it is the largest possible subgraph: you could not find another node anywhere in the graph such that it could be added to the subgraph and all the nodes in the subgraph would still be connected.

### What are independent sets in graph theory?

An independent set of edges is a set of edges of which no two have a vertex in common. It is usually called a matching. A vertex coloring is a partition of the vertex set into independent sets.

### How do you find an independent set?

Typical way to find independent sets is to consider the complement of a graph. A complement of a graph is defined as a graph with the same set of vertices and an edge between a pair if and only if there is no edge between them in the original graph.

What is an independent set in a graph?

In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set of vertices such that for every two vertices in , there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in.

What is independent set in graph theory?

## What’s the difference between maximum and maximal?

‘Maximum’ is the greatest; ‘maximal’ is tending towards the greatest.

## What is a linearly independent set?

An infinite set of vectors is linearly independent if every nonempty finite subset is linearly independent. Otherwise, the family is said linearly dependent. A set of vectors which is linearly independent and spans some vector space, forms a basis for that vector space.

How many sets are independent?

Every graph contains at most 3n/3 maximal independent sets, but many graphs have far fewer. The number of maximal independent sets in n-vertex cycle graphs is given by the Perrin numbers, and the number of maximal independent sets in n-vertex path graphs is given by the Padovan sequence.

Where do you graph independent and dependent variables?

The independent variable belongs on the x-axis (horizontal line) of the graph and the dependent variable belongs on the y-axis (vertical line).

### How many maximal independent sets are there in a graph?

The graph of the cube has six different maximal independent sets (two of them are maximum), shown as the red vertices. In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set.

### When is a maximal independent set a dominating set?

Every maximal independent set is a dominating set, a set of vertices such that every vertex in the graph either belongs to the set or is adjacent to the set. A set of vertices is a maximal independent set if and only if it is an independent dominating 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.

How is a maximal independent graph of a Turan graph formed?

Any maximal independent set in this graph is formed by choosing one vertex from each triangle. The complementary graph, with exactly 3 n/3 maximal cliques, is a special type of Turán graph; because of their connection with Moon and Moser’s bound, these graphs are also sometimes called Moon-Moser graphs.