Line/Edge Covering

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, it has a minimum degree of 1.

Example

Take a look at the following graph ?

Its subgraphs having line covering are ?

C1 = {{a, b}, {c, d}}
C2 = {{a, d}, {b, c}}
C3 = {{a, b}, {b, c}, {b, d}}
C4 = {{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 without leaving some vertex uncovered.

From the above coverings − C₁, C₂, C₃ are minimal line coverings, while C₄ is not because we can delete {b, c} and all vertices remain covered.

Minimum Line Covering

A minimal line covering with the minimum number of edges is called a minimum line covering of 'G'. The number of edges in a minimum line covering is called the line covering number of 'G' (α₁).

In the above example, C₁ and C₂ are the minimum line coverings of G and α₁ = 2.

Key Properties

• Every line covering contains a minimal line covering.
• Every line covering does not necessarily contain a minimum line covering (C₃ 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 components of 'C' are star graphs, and no edge can be deleted from a star graph.

Conclusion

A line covering is a set of edges that touches every vertex in the graph. The minimum line covering uses the fewest edges possible (⌈n/2⌉ for n vertices), and its size is called the line covering number α₁.