3. Consider a formula Vr3y3zVuE(x,y) ^ E(x, z) ^ (u = x v ¬E(u, y) V ¬E(u, z)). Here, x, y, z, u E V for some graph G = (V, E), and E() is the edge relation. Assume that Vu¬E(v, v), and that |V| = n. (a) What is the maximal possible number of edges in an undirected graph satisfying this formula? (b) What is the minimal possible number of edges in an directed graph satisfying this formula? Show both that there exists a graph with that many edges (you can just describe how to construct such a graph), and that this number is optimal.
3. Consider a formula Vr3y3zVuE(x,y) ^ E(x, z) ^ (u = x v ¬E(u, y) V ¬E(u, z)). Here, x, y, z, u E V for some graph G = (V, E), and E() is the edge relation. Assume that Vu¬E(v, v), and that |V| = n. (a) What is the maximal possible number of edges in an undirected graph satisfying this formula? (b) What is the minimal possible number of edges in an directed graph satisfying this formula? Show both that there exists a graph with that many edges (you can just describe how to construct such a graph), and that this number is optimal.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:3. Consider a formula Væ3y3zVuE(x, y) ^ E(x, z) ^ (u = x V ¬E(u, y) V ¬E(u, z)). Here, 2, y, z, u e V
for some graph G = (V, E), and E() is the edge relation. Assume that Vu¬E(v, v), and that |V| = n.
(a) What is the maximal possible number of edges in an undirected graph satisfying this formula?
(b) What is the minimal possible number of edges in an directed graph satisfying this formula?
Show both that there exists a graph with that many edges (you can just describe how to construct
such a graph), and that this number is optimal.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
