Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 418 — #426
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Chapter 10
Directed graphs & Partial Orders
Problem 10.45.
In an n-player round-robin tournament, every pair of distinct players compete in a
single game. Assume that every game has a winner—there are no ties. The results
of such a tournament can then be represented with a tournament digraph where the
vertices correspond to players and there is an edge hx !yi iff x beat y in their
game.
(a) Explain why a tournament digraph cannot have cycles of length one or two.
(b) Is the “beats” relation for a tournament graph always/sometimes/never:
asymmetric?
reflexive?
irreflexive?
transitive?
Explain.
(c) Show that a tournament graph is a linear order iff there are no cycles of length
three.
Homework Problems
Problem 10.46.
Let R and S be transitive binary relations on the same set A. Which of the following
new relations must also be transitive? For each part, justify your answer with a brief
argument if the new relation is transitive and a counterexample if it is not.
(a) R 1
(b) R \ S
(c) R ı R
(d) R ı S
Exam Problems
Problem 10.47.
Suppose the precedence constraints on a set of 32 unit time tasks was isomorphic
to the powerset, pow.f1; 2; 3; 4; 5g/ under the strict subset relation .
For example, the task corresponding to the set f2; 4g must be completed before
the task corresponding to the set f1; 2; 4g because f2; 4g f1; 2; 4g; the task