Graph Theory with Applications to Engineering & Computer Science by Narsingh Deo

Author:Narsingh Deo [Narsingh Deo]
Language: eng
Format: epub, mobi
Publisher: Dover Publications, Inc
Published: 2017-01-18T05:00:00+00:00

Show with this ordering of vertices in R(G) that digraph G is acyclic if and only if R(G) is an upper triangular matrix.

9-24.Prove that a digraph G is acyclic if and only if every element on the principal diagonal of its reachability (or accessibility) matrix R(G) is zero. 9-25. Prove that an acyclic digraph G of n vertices has a unique directed Hamiltonian path if and only if the number of nonzero elements in R(G) is n(n − l)/2.

9-26.There are 15 computer programs that must be processed according to the following set of orders :

1 > 2,7,13,

2 > 3, 8,14,

3 > 9,15,

4>3,

5>4,11,

6 > 5,12,

7>6,

8 > 7, 9,14,

9> 15,

10 > 4, 9,

11 > 10,

12 > 11,

13 > 7, 12,

14 > 13,15,

where 1 > 2, 7, 13 means that programs 2, 7, and 13 can be processed only after program 1 has been processed. Is it possible for the programs to be processed? If so, give a processing sequence. [Hint: Write X(G); derive R(C) from X(G) using Problem 9-11. Use Problem 9-23 to check if G is acyclic]

9-27.A digraph defined on the relation “is a parent of” is called a genetic digraph. (Genetic digraphs are useful in biology.) Investigate the properties of genetic digraphs.

9-28.Use digraphs to solve the classical problem of “three cannibals and three edible missionaries seeking to cross a river in a boat that can hold at most two people, and all the missionaries and one of the cannibals can row the boat. Also, at no time should the cannibals outnumber the missionaries on either shore.” (Hint: Represent each state by a vertex and a possible transition by a directed edge.)

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