# Graph Theory - Coverings

A covering graph is a subgraph which contains either all the vertices or all the edges corresponding to some other graph. A subgraph which contains all the vertices is called a **line/edge covering**. A subgraph which contains all the edges is called a **vertex covering**.

## Line Covering

Let G = (V, E) be a graph. A subset C(E) is called a line covering of G if every vertex of G is incident with at least one edge in C, i.e.,

deg(V) ≥ 1 ∀ V ∈ G

because each vertex is connected with another vertex by an edge. Hence it has a minimum degree of 1.

### Example

Take a look at the following graph −

Its subgraphs having line covering are as follows −

C_{1}= {{a, b}, {c, d}} C_{2}= {{a, d}, {b, c}} C_{3}= {{a, b}, {b, c}, {b, d}} C_{4}= {{a, b}, {b, c}, {c, d}}

Line covering of ‘G’ does not exist if and only if ‘G’ has an isolated vertex. Line covering of a graph with ‘n’ vertices has at least ⌊ n 2 ⌋ edges.

## Minimal Line Covering

A line covering C of a graph G is said to be minimal **if no edge can be deleted from C**.

### Example

In the above graph, the subgraphs having line covering are as follows −

C_{1}= {{a, b}, {c, d}} C_{2}= {{a, d}, {b, c}} C_{3}= {{a, b}, {b, c}, {b, d}} C_{4}= {{a, b}, {b, c}, {c, d}}

Here, C_{1}, C_{2}, C_{3} are minimal line coverings, while C_{4} is not because we can delete {b, c}.

## Minimum Line Covering

It is also known as **Smallest Minimal Line Covering**. A minimal line covering with minimum number of edges is called a minimum line covering of ‘G’. The number of edges in a minimum line covering in ‘G’ is called the **line covering number** of ‘G’ (α_{1}).

### Example

In the above example, C_{1} and C_{2} are the minimum line covering of G and α_{1} = 2.

Every line covering contains a minimal line covering.

Every line covering does not contain a minimum line covering (C

_{3}does not contain any minimum line covering.No minimal line covering contains a cycle.

If a line covering ‘C’ contains no paths of length 3 or more, then ‘C’ is a minimal line covering because all the components of ‘C’ are star graph and from a star graph, no edge can be deleted.

## Vertex Covering

Let ‘G’ = (V, E) be a graph. A subset K of V is called a vertex covering of ‘G’, if every edge of ‘G’ is incident with or covered by a vertex in ‘K’.

### Example

Take a look at the following graph −

The subgraphs that can be derived from the above graph are as follows −

K_{1}= {b, c} K_{2}= {a, b, c} K_{3}= {b, c, d} K_{4}= {a, d}

Here, K_{1}, K_{2}, and K_{3} have vertex covering, whereas K_{4} does not have any vertex covering as it does not cover the edge {bc}.

## Minimal Vertex Covering

A vertex ‘K’ of graph ‘G’ is said to be minimal vertex covering if no vertex can be deleted from ‘K’.

### Example

In the above graph, the subgraphs having vertex covering are as follows −

K_{1} = {b, c}

K_{2} = {a, b, c}

K_{3} = {b, c, d}

Here, K_{1} and K_{2} are minimal vertex coverings, whereas in K_{3}, vertex ‘d’ can be deleted.

## Minimum Vertex Covering

It is also known as the smallest minimal vertex covering. A minimal vertex covering of graph ‘G’ with minimum number of vertices is called the minimum vertex covering.

The number of vertices in a minimum vertex covering of ‘G’ is called the vertex covering number of G (α_{2}).

### Example

In the following graph, the subgraphs having vertex covering are as follows −

K_{1} = {b, c}

K_{2} = {a, b, c}

K_{3} = {b, c, d}

Here, K_{1} is a minimum vertex cover of G, as it has only two vertices. α_{2} = 2.