Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 400 — #408
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Chapter 10
Directed graphs & Partial Orders
(c) Prove that every finite tournament digraph has a ranking.
Optional
(d) Prove that the greater-than relation > on the rational numbers Q is a DAG and
a tournament graph that has no ranking.
Homework Problems
Problem 10.11.
Let R be a binary relation on a set A. Regarding R as a digraph, let W .n/ denote
the length-n walk relation in the digraph R, that is,
(a) Prove that
a W .n/ b WWD there is a length n walk from a to b in R:
W .n/ ı W .m/ D W .mCn/ (10.11)
for all m; n 2 N, where ı denotes relational composition.
(b) Let R n be the composition of R with itself n times for n 0. So R 0 WWD Id A ,
and R nC1 WWD R ı R n .
Conclude that
for all n 2 N.
(c) Conclude that
R n D W .n/ (10.12)
R C D
jAj
[
where R C is the positive length walk relation determined by R on the set A.
iD1
R i
Problem 10.12.
We can represent a relation S between two sets A D fa 1 ; : : : ; a n g and B D
fb 1 ; : : : ; b m g as an n m matrix M S of zeroes and ones, with the elements of
M S defined by the rule
M S .i; j / D 1 IFF a i S b j :
If we represent relations as matrices this way, then we can compute the composition
of two relations R and S by a “boolean” matrix multiplication ˝ of their