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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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.
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.
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