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Questions and Answers of Statistics

In Exercise use Kuratowski's theorem to determine whether the given graph is planar.
Find the crossing numbers of each of these non planar graphs. a) K5 b) K6 c) K7 d) K3,4 e) K4,4 f) K5,5
Show that if m and n are even positive integers, the crossing number of Km,n is less than or equal to mn(m − 2) (n − 2)/16.
In Exercise draw the given planar graph without any crossings.
Find the thickness of the graphs in Exercise 27. In Exercise 27 a) K5 b) K6 c) K7 d) K3,4 e) K4,4 f) K5,5
Use Exercise 32 to show that the thickness of Kn is at least [(n + 7)/6] whenever n is a positive integer.
DrawK3,3 on the surface of a torus so that no edges cross.
In Exercise determine whether the given graph is planar. If so, draw it so that no edges cross.
In Exercise construct the dual graph for the map shown. Then find the number of colors needed to color the map so that no two adjacent regions have the same color
Which graphs have a chromatic number of 1?
What is the chromatic number of Wn?
Schedule the final exams for Math 115, Math 116, Math 185, Math 195, CS 101, CS 102, CS 273, and CS 473, using the fewest number of different time slots, if there are no students taking both Math 115
The mathematics department has six committees, each meeting once a month. Howmany different meeting times must be used to ensure that no member is scheduled to attend two meetings at the same time if
Find the edge chromatic numbers of a) Cn, where n ≥ 3. b) Wn, where n ≥ 3.
Seven variables occur in a loop of a computer program. The variables and the steps during which they must be stored are t: steps 1 through 6; u: step 2; v: steps 2 through 4; w: steps 1, 3, and 5; x:
Construct a coloring of the graph shown using this algorithm.
Show that the coloring produced by this algorithm may use more colors than are necessary to color a graph. A connected graph G is called chromatically k-critical if the chromatic number of G is k,
Show that Wn is chromatically 4-critical whenever n is an odd integer, n ≥ 3.
Show that if G is a chromatically k-critical graph, then the degree of every vertex of G is at least k − 1.
Frequencies for mobile radio (or cellular) telephones are assigned by zones. Each zone is assigned a set of frequencies to be used by vehicles in that zone. The same frequency cannot be used in
Show that every planar graph G can be colored using five or fewer colors
Show that g(5) = 1. That is, show that all pentagons can be guarded using one point.
Show that g(n) ‰¥ [n/3]. [Consider the polygon with 3k vertices that resembles a comb with k prongs, such as the polygon with 15 sides shown here.]
In Exercise find the chromatic number of the given graph.
Which of these graphs are trees?a)b) c) d) e) f)
a) How many nonisomorphic unrooted trees are there with three vertices? b) How many nonisomorphic rooted trees are there with three vertices (using isomorphism for directed graphs)?
a) How many nonisomorphic unrooted trees are there with five vertices? b) How many nonisomorphic rooted trees are there with five vertices (using isomorphism for directed graphs)?
Let G be a simple graph with n vertices. Show that a) G is a tree if and only if it is connected and has n - 1 edges. b) G is a tree if and only if G has no simple circuits and has n − 1 edges.
How many edges does a full binary tree with 1000 internal vertices have?
Suppose 1000 people enter a chess tournament. Use a rooted tree model of the tournament to determine how many games must be played to determine a champion, if a player is eliminated after one loss
A chain letter starts with a person sending a letter out to 10 others. Each person is asked to send the letter out to 10 others, and each letter contains a list of the previous six people in the
Either draw a full m-ary tree with 84 leaves and height 3, where m is a positive integer, or show that no such tree exists.
Construct a complete binary tree of height 4 and a complete 3-ary tree of height 3.
Prove a) Part (ii) of Theorem 4. b) Part (iii) of Theorem 4.
Answer these questions about the rooted tree illustrated.a) Which vertex is the root? b) Which vertices are internal? c) Which vertices are leaves? d) Which vertices are children of j? e) Which
How many edges are there in a forest of t trees containing a total of n vertices?
How many different isomers do these saturated hydrocarbons have? a) C3H8 b) C5H12 c) C6H14
Answer the same questions as those given in Exercise 34 for a rooted tree representing a computer file system. a) The parent of a vertex b) A child of a vertex c) A sibling of a vertex d) The
Let n be a power of 2. Show that n numbers can be added in log n steps using a tree-connected network of n - 1 processors.
The eccentricity of a vertex in an unrooted tree is the length of the longest simple path beginning at this vertex.A vertex is called a center if no vertex in the tree has smaller eccentricity than
Show that a tree has either one center or two centers that are adjacent.
The rooted Fibonacci trees Tn are defined recursively in the following way. T1 and T2 are both the rooted tree consisting of a single vertex, and for n = 3, 4, . . . , the rooted tree Tn is
What is wrong with the following "proof" using mathematical induction of the statement that every tree with n vertices has a path of length n − 1. Basis step: Every tree with one vertex clearly has
Answer the same questions as listed in Exercise 3 for the rooted tree illustrated.
What is the level of each vertex of the rooted tree in Exercise 3?
Draw the subtree of the tree in Exercise 3 that is rooted at a) a. b) c. c) e.
Build a binary search tree for the words banana, peach, apple, pear, coconut, mango, and papaya using alphabetical order.
Find the least number of comparisons needed to sort four elements and devise an algorithm that sorts these elements using this number of comparisons.
Complete the tournament sort of the list 22, 8, 14, 17, 3, 9, 27, 11. Show the labels of the vertices at each step.The tournament sort is a sorting algorithm that works by building an ordered binary
Describe the tournament sort using pseudocode.The tournament sort is a sorting algorithm that works by building an ordered binary tree. We represent the elements to be sorted by vertices that will
How many comparisons does the tournament sort use to find the second largest, the third largest, and so on, up to the (n − 1)st largest (or second smallest) element?
Which of these codes are prefix codes? a) a: 11, e: 00, t : 10, s: 01 b) a: 0, e: 1, t : 01, s: 001 c) a: 101, e: 11, t : 001, s: 011, n: 010 d) a: 010, e: 11, t : 011, s: 1011, n: 1001, i: 10101
What are the codes for a, e, i, k, o, p, and u if the coding scheme is represented by this tree?
Use Huffman coding to encode these symbols with given frequencies: a: 0.20, b: 0.10, c: 0.15, d: 0.25, e: 0.30. What is the average number of bits required to encode a character?
Construct two different Huffman codes for these symbols and frequencies: t : 0.2, u: 0.3, v: 0.2, w: 0.3.
Construct a Huffman code for the letters of the English alphabet where the frequencies of letters in typical English text are as shown in this table.Suppose that m is a positive integer with m
Using the symbols 0, 1, and 2 use ternary (m = 3) Huffman coding to encode these letters with the given frequencies: A: 0.25, E: 0.30, N: 0.10, R: 0.05, T: 0.12, Z: 0.18.
How many comparisons are needed to locate or to add each of thesewords in the search tree for Exercise 1, starting fresh each time? a) Pear b) Banana c) Kumquat d) Orange
Given n + 1 symbols x1, x2, . . . , xn, xn+1 appearing 1, f1, f2, . . . , fn times in a symbol string, respectively, where fj is the j th Fibonacci number, what is the maximum number of bits used to
Draw a game tree for nim if the starting position consists of two piles with two and three stones, respectively. When drawing the tree represent by the same vertex symmetric positions that result
Suppose that we vary the payoff to the winning player in the game of nim so that the payoff is n dollars when n is the number of legal moves made before a terminal position is reached. Find the
Draw the subtree of the game tree for tic-tac-toe beginning at each of these positions. Determine the value of each of these subtrees.a)b) c) d)
Show that if a game of nim begins with two piles containing the same number of stones, as long as this number is at least two, then the second player wins when both players follow optimal strategies.
How many children does the root of the game tree for checkers have? How many grandchildren does it have?
Draw the game tree for the game of tic-tac-toe for the levels corresponding to the first two moves. Assign the value of the evaluation function mentioned in the text that assigns to a position the
Using alphabetical order, construct a binary search tree for the words in the sentence "The quick brown fox jumps over the lazy dog."
How many weighings of a balance scale are needed to find a counterfeit coin among four coins if the counterfeit coin may be either heavier or lighter than the others? Describe an algorithm to find
How many weighings of a balance scale are needed to find a counterfeit coin among 12 coins if the counterfeit coin is lighter than the others? Describe an algorithm to find the lighter coin using
In Exercises 1-2 construct the universal address system for the given ordered rooted tree. Then use this to order its vertices using the lexicographic order of their labels.1.2.
In which order are the vertices of the ordered rooted tree in Exercise 8 visited using an inorder traversal?
In which order are the vertices of the ordered rooted tree in Exercise 7 visited using a postorder traversal?
In which order are the vertices of the ordered rooted tree in Exercise 9 visited using a postorder traversal?
a) Represent the expressions (x + xy) + (x/y) and x + ((xy + x)/y) using binary trees. Write these expressions inb) Prefix notation.c) Postfix notation.d) Infix notation.
a) Represent (A ∩ B) − (A ∪ (B − A)) using an ordered rooted tree.Write this expression inb) Prefix notation.c) Postfix notation.d) Infix notation.
In how many ways can the string A ∩ B − A ∩ B − A be fully parenthesized to yield an infix expression?
What is the value of each of these prefix expressions? a) − ∗ 2 / 8 4 3 b) ↑ − ∗ 3 3 ∗ 4 2 5 c) + − ↑ 3 2 ↑ 2 3 / 6 − 4 2 d) ∗ + 3 + 3 ↑ 3 + 3 3 3
Construct the ordered rooted tree whose preorder traversal is a, b, f, c, g, h, i, d, e, j, k, l, where a has four children, c has three children, j has two children, b and e have one child each, and
Show that an ordered rooted tree is uniquely determined when a list of vertices generated by a postorder traversal of the tree and the number of children of each vertex are specified.
Show that any well-formed formula in prefix notation over a set of symbols and a set of binary operators contains exactly one more symbol than the number of operators.
Give six examples of well-formed formulae with three or more operators in postfix notation over the set of symbols {x, y, z} and the set of operators {+,×, ◦}.
Suppose that the vertex with the largest address in an ordered rooted tree T has address 2.3.4.3.1. Is it possible to determine the number of vertices in T?
Determine the order in which a preorder traversal visits the vertices of the given ordered rooted tree.
How many different spanning trees does each of these simple graphs have? a) K3 b) K4 c) K2,2 d) C5
In Exercises 1-2 use depth-first search to produce a spanning tree for the given simple graph. Choose a as the root of this spanning tree and assume that the vertices are ordered alphabetically.1.2.
Use depth-first search to find a spanning tree of each of these graphs. a) W6 (see Example 7 of Section 10.2), starting at the vertex of degree 6 b) K5 c) K3,4, starting at a vertex of degree 3 d) Q3
Describe the trees produced by breadth-first search and depth-first search of the wheel graph Wn, starting at the vertex of degree n, where n is an integer with n ≥ 3. (See Example 7 of Section
Describe the trees produced by breadth-first search and depth-first search of the complete bipartite graph Km,n, starting at a vertex of degreem, wheremand n are positive integers. Justify your
Suppose that an airline must reduce its flight schedule to save money. If its original routes are as illustrated here, which flights can be discontinued to retain service between all pairs of cities
Show that the length of the shortest path between vertices v and u in a connected simple graph equals the level number of u in the breadth-first spanning tree of G with root v.
Use backtracking to solve the n-queens problem for these values of n. a) n = 3 b) n = 5 c) n = 6
Explain how backtracking can be used to find a Hamilton path or circuit in a graph.
Find a spanning tree for the graph shown by removing edges in simple circuits.
Show that every finite simple graph has a spanning forest. A spanning forest of a graph G is a forest that contains every vertex of G such that two vertices are in the same tree of the forest when
How many edges must be removed to produce the spanning forest of a graph with n vertices, m edges, and c connected components?
Explain how to use breadth-first search to find the length of a shortest path between two vertices in an undirected graph.
Devise an algorithm based on breadth-first search for finding the connected components of a graph.
Which connected simple graphs have exactly one spanning tree?
Devise an algorithm for constructing the spanning forest of a graph based on depth-first searching.
Let G be a connected graph. Show that if T is a spanning tree of G constructed using depth-first search, then an edge of G not in T must be a back edge, that is, it must connect a vertex to one of
For which graphs do depth-first search and breadth-first search produce identical spanning trees no matter which vertex is selected as the root of the tree? Justify your answer.

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