Finding the line covering number of a graph

The line covering number (also called the edge cover number) of a graph is the minimum number of edges required to cover all the vertices of the graph. An edge cover is a set of edges such that every vertex in the graph is an endpoint of at least one edge in the set.

The line covering number is denoted by α₁.

Lower Bound Formula

For a graph with n vertices, the line covering number has the following lower bound −

α₁ ≥ ⌈n / 2⌉

This is because each edge can cover at most 2 vertices. So to cover all n vertices, you need at least ⌈n / 2⌉ edges.

Problem Statement

What is the line covering number for the following graph ?

Solution

The graph has the following properties −

Number of vertices = |V| = n = 7

Using the lower bound formula −

α? ≥ ⌈n / 2⌉
α? ≥ ⌈7 / 2⌉
α? ≥ ⌈3.5⌉
α? ≥ 4

By examining the graph, we can find an edge cover that uses exactly 4 edges to cover all 7 vertices. The following diagram highlights one such minimum edge cover −

The four cover edges and the vertices they cover are −

All 7 vertices are covered. Since we cannot cover 7 vertices with fewer than 4 edges (3 edges cover at most 6 vertices), the line covering number is −

α₁ = 4

Conclusion

The line covering number of a graph is the minimum number of edges needed to cover all vertices. It is always at least ⌈n / 2⌉, where n is the number of vertices. To find it, apply the lower bound formula and then verify by identifying an edge cover of that size in the graph.