Alpha(G) and omega(G) for a subgraph G of a graph H
Not what you're looking for?
The stability number, alpha(G), of a graph G is the cardinality of the largest subset S of V(G), the vertex set of G, such that no two of the vertices in S are connected by an edge of G.
The clique number, omega(G), of a graph G is the cardinality of the largest subset S of V(G), the vertex set of G, such that every pair of vertices in S are connected by an edge of G.
Let G, H be graphs such that G is a subgraph of H. Prove or disprove each of the following:
(a) alpha(G) <= alpha(H)
(b) alpha(G) >= alpha(H)
(c) omega(G) <= omega(H)
(d) omega(G) >= omega(H)
Purchase this Solution
Solution Summary
Step-by-step proofs are given for the given statements that are true. Counterexamples (with detailed justifications) for the given statements that are false are provided in an attached .doc file.
Solution Preview
Statement (a) is true.
Proof of (a):
Since G is a subgraph of H, every pair of vertices v1, v2 of G that are not connected by an edge in G are also not connected by an edge in H.
Thus every subset S of V(G), the vertex set of G, such that no two of the vertices in S are connected by an edge of G is also a subset of V(H), the vertex set of H, such that no two of the vertices in S are connected by an edge of H.
Hence the cardinality of the largest subset S of V(G) such that no two of the vertices in S are ...
Education
- AB, Hood College
- PhD, The Catholic University of America
- PhD, The University of Maryland at College Park
Recent Feedback
- "Thanks for your assistance. "
- "Thank you. I understand now."
- "Super - Thank You"
- "Very clear. I appreciate your help. Thank you."
- "Great. thank you so much!"
Purchase this Solution
Free BrainMass Quizzes
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.