Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies
- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who
Independent Line Set
Independent sets are represented in sets, in which
there should not be any edges adjacent to each other. There should not be any common vertex between any two edges.
there should not be any vertices adjacent to each other. There should not be any common edge between any two vertices.
Independent Line Set
Let 'G' = (V, E) be a graph. A subset L of E is called an independent line set of 'G' if no two edges in L are adjacent. Such a set is called an independent line set.
Example
Let us consider the following subsets −
L1 = {a,b}
L2 = {a,b} {c,e}
L3 = {a,d} {b,c}In this example, the subsets L2 and L3 are clearly not the adjacent edges in the given graph. They are independent line sets. However L1 is not an independent line set, as for making an independent line set, there should be at least two edges.
Maximal Independent Line Set
An independent line set is said to be the maximal independent line set of a graph 'G' if no other edge of 'G' can be added to 'L'.
Example
Let us consider the following subsets −
L1 = {a, b}
L2 = {{b, e}, {c, f}}
L3 = {{a, e}, {b, c}, {d, f}}
L4 = {{a, b}, {c, f}}L2 and L3 are maximal independent line sets/maximal matching. As for only these two subsets, there is no chance of adding any other edge which is not an adjacent. Hence these two subsets are considered as the maximal independent line sets.
Maximum Independent Line Set
A maximum independent line set of 'G' with maximum number of edges is called a maximum independent line set of 'G'.
Number of edges in a maximum independent line set of G (β1) = Line independent number of G = Matching number of G
Example
Let us consider the following subsets −
L1 = {a, b}
L2 = {{b, e}, {c, f}}
L3 = {{a, e}, {b, c}, {d, f}}
L4 = {{a, b}, {c, f}}L3 is the maximum independent line set of G with maximum edges which are not the adjacent edges in graph and is denoted by β1 = 3.
Note − For any graph G with no isolated vertex,
α1 + β1 = number of vertices in a graph = |V|
Example
Line covering number of Kn/Cn/wn,
Line independent number (Matching number) = β1 = ⌊ n 2 ⌋ α1 + β1 = n
190 Views
