Independent Line Set

An independent set in a graph is a set of elements (vertices or edges) where no two elements are adjacent to each other. There are two types −

• Independent line set (edge independent set) − A set of edges where no two edges share a common vertex.
• Independent vertex set − A set of vertices where no two vertices share a common edge.

Independent Line Set

Let G = (V, E) be a graph. A subset L of E is called an independent line set (also called a matching) if no two edges in L are adjacent − that is, no two edges share a common vertex.

Example

Consider the following subsets −

L? = { {a,b} }                    ? Only 1 edge (trivially independent)
L? = { {a,b}, {c,e} }             ? Independent (no shared vertex) ?
L? = { {a,d}, {b,c} }             ? Independent (no shared vertex) ?

L₂ and L₃ are independent line sets because no two edges in each set share a common vertex.

Maximal Independent Line Set

An independent line set L is maximal if no other edge of G can be added to L without making two edges adjacent. In other words, every edge not in L shares a vertex with at least one edge already in L.

Example

Consider the following subsets −

L? = { {a,b} }                          ? Not maximal (can add more edges)
L? = { {b,e}, {c,f} }                   ? Maximal (no more edges can be added) ?
L? = { {a,e}, {b,c}, {d,f} }            ? Maximal (no more edges can be added) ?
L? = { {a,b}, {c,f} }                   ? Not maximal (can add {d,g})

L₂ and L₃ are maximal independent line sets because no additional edge can be included without creating an adjacency.

Maximum Independent Line Set

A maximum independent line set is the maximal independent line set with the largest number of edges. The number of edges in this set is called the matching number of G, denoted by β₁.

From the subsets above:
L? has 2 edges
L? has 3 edges  ? Maximum

?? = 3  (matching number)

L₃ = { {a,e}, {b,c}, {d,f} } is the maximum independent line set with β₁ = 3.

Gallai's Theorem

For any graph G with no isolated vertex, the line covering number (α₁) and matching number (β₁) satisfy −

α₁ + β₁ = |V|

For common graph families −

For K?, C?, and W?:
    ?? = ?n / 2?
    ?? = n - ?n / 2? = ?n / 2?
    ?? + ?? = n  ?

Conclusion

An independent line set (matching) is a set of non-adjacent edges. A maximal matching cannot have any more edges added, and a maximum matching has the largest possible size (β₁). Together with the line covering number, they satisfy Gallai's theorem: α₁ + β₁ = |V|.