Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 399 — #407
10.11. Summary of Relational Properties 399
Theorem. If W is the minimum weight matrix for length k walks in a weighted
graph G, and V is the minimum weight matrix for length m walks, then W V is
min+
the minimum weight matrix for length k C m walks.
Problems for Section 10.4
Practice Problems
Problem 10.9.
Let
A WWD f1; 2; 3g
B WWD f4; 5; 6g
R WWD f.1; 4/; .1; 5/; .2; 5/; .3; 6/g
S WWD f.4; 5/; .4; 6/; .5; 4/g:
Note that R is a relation from A to B and S is a relation from B to B.
List the pairs in each of the relations below.
(a) S ı R.
(b) S ı S.
(c) S 1 ı R.
Problem 10.10.
In a round-robin tournament, every two distinct players play against each other
just once. For a round-robin tournament with no tied games, a record of who beat
whom can be described 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 ranking is a path that includes all the players. So in a ranking, each player won
the game against the next lowest ranked player, but may very well have lost their
games against much lower ranked players—whoever does the ranking may have a
lot of room to play favorites.
(a) Give an example of a tournament digraph with more than one ranking.
(b) Prove that if a tournament digraph is a DAG, then it has at most one ranking.