Show the following problem is in NP (nondeterministic polynomial time):
Homework Answers
Let the input x be two graphs G1 and G2 and let y be the indices {i1,i2,···,in}.
An algorithm A(x,y) verifies that the graphs are isomorphic by executing the following steps:
(a) Check if y is a permutation of {1,2,···,n}.
If no, return false; else continue.
(b) Permute the vertices of G1 as given by the given permutation. Verify that the permuted G1 is identical to G2.
Step (a) takes at most O(V^2) time and step (b) runs in O(V+E) time, therefore the verification algorithm A runs in O(V*2) time. Hence the problem is NP-complete.
Add Answer to:
Show the following problem is in NP (nondeterministic polynomial
time):
Given two graphs G1 = (V1,...
Exercise 6. Given two graphs Gi and G2, consider the graph G1DG2 constructed as follows: the...
Exercise 6. Given two graphs Gi and G2, consider the graph G1DG2 constructed as follows: the vertices of GIG2 are the pairs (v1, v2), where 1 is a vertex of G1 and v2 is a vertex of G2 two vertices (u1, u2) and (v1, v2) in GIG2 are joined by edge whenever (u1 is adjacent to v2 in G2) or (u1 is adjacent to vi in G1, and u2 (i) Show the following: if G1 and G2 are connected, then...
Discrete math. Question 4.(10+16=26 points) Let G = (V1, E1) and H = (V2, E2) be...
Discrete math.
Question 4.(10+16=26 points) Let G = (V1, E1) and H = (V2, E2) be the following graphs: a с u V b z W e d X G = (V1, E1) y H = (V2, E2) a) Draw the complement G of G. b) Show that G and H are isomorphic by writing a graph isomorphism F : V1 + V2.
19. Use the definition below and the minimum criteria a graph must meet in order to be potentially isomorphic to answer the question Recall Definition: Isomorphism: For a graph G1 V, E13 and G2-V2, E...
19. Use the definition below and the minimum criteria a graph must meet in order to be potentially isomorphic to answer the question Recall Definition: Isomorphism: For a graph G1 V, E13 and G2-V2, E23 G1is isomorphic to G2 denoted GG2 iff af:V V2where i) f is bijective and Describe an isomorphism between the following two graphs, or briefly explain why no such isomorphism exists. f(A)1 f(B)6 f(C) 3 f(D)8 IG)2 f(H)5 f(I)4
19. Use the definition below and the...
Graphs (15 points) 14. For the following graph (8 points): a. Find all the edges that...
Graphs (15 points) 14. For the following graph (8 points): a. Find all the edges that are incident of v1: b. Find all the vertices that are adjacent to v3: C. Find all the edges that are adjacent to e1: d. Find all the loops: e. Find all the parallel edges: f. Find all the isolated vertices: g. Find the degree of v3: h. Find the total degree of the graph: e3 e2 V2 VI 26 e4 e7 es 05...
Lemma. If two vector spaces have the same dimension then they are isomorphic Proof. To show that ...
Lemma. If two vector spaces have the same dimension then they are isomorphic Proof. To show that any two spaces of dimension n are isomorphic, we can simply show that any one is isomorphic to R. Then we will have shown that they are isomorphic to each other, by the transitivity of isomorphism (which was established in the first Theorem of this section) Theorem 1 Isomorphism is an equivalence relation among ctor spaces Let v be n--dimensional. Fix a basis...
Please answer part C in detail: Problem 1: The following two NFAs, G1 and G2, represent...
Please answer part C in detail:
Problem 1: The following two NFAs, G1 and G2, represent behaviors of a certain discrete-event system. In this question, you are expected to show steps of your construction method. You may use software to help you with your construction but a single screenshot of the end result, without any explanation or construction steps, will not be accepted. G1: Marks sequences that end with a suffix bbc G2: Marks sequences that end with a suffix...
Problem 5. (12 marks) Connectivity in undirected graphs vs. directed graphs. a. (8 marks) Prove that...
Problem 5. (12 marks) Connectivity in undirected graphs vs. directed graphs. a. (8 marks) Prove that in any connected undirected graph G- (V, E) with VI > 2, there are at least two vertices u, u є V whose removal (along with all the edges that touch them) leaves G still connected. Propose an efficient algorithm to find two such vertices. (Hint: The algorithm should be based on the proof or the proof should be based on the algorithm.) b....
Please do only e and f and show work null(AT) null(A) T col(A) row(A) Figure 5.6 The four fundamental subspaces (f) Fin...
Please do only e and f and show work
null(AT) null(A) T col(A) row(A) Figure 5.6 The four fundamental subspaces (f) Find bases for the four fundamental subspaces of 1 1 1 6 -1 0 1 -1 2 A= -2 3 1 -2 1 4 1 6 1 3 8. Given a subspace W of R", define the orthogonal complement of W to be W vE R u v 0 for every u E W (a) Let W span(e, e2)...
Show that the following problem is NP-Complete (Hint: reduce from 3-SAT or Vertex Cover). Given an undirected graph G wi...
Show that the following problem is NP-Complete (Hint: reduce from 3-SAT or Vertex Cover). Given an undirected graph G with positive integer distances on the edges, and two integers f and d, is there a way to select f vertices on G on which to locate firehouses, so that no vertex of G is at distance more than d from a firehouse?
(a) Given a graph G = (V, E) and a number k (1 ≤ k ≤...
(a) Given a graph G = (V, E) and a number k (1 ≤ k ≤ n), the CLIQUE problem asks us whether there is a set of k vertices in G that are all connected to one another. That is, each vertex in the ”clique” is connected to the other k − 1 vertices in the clique; this set of vertices is referred to as a ”k-clique.” Show that this problem is in class NP (verifiable in polynomial time)...
Exercise 6. Given two graphs Gi and G2, consider the graph G1DG2 constructed as follows: the vertices of GIG2 are the pairs (v1, v2), where 1 is a vertex of G1 and v2 is a vertex of G2 two vertices (u1, u2) and (v1, v2) in GIG2 are joined by edge whenever (u1 is adjacent to v2 in G2) or (u1 is adjacent to vi in G1, and u2 (i) Show the following: if G1 and G2 are connected, then...
Discrete math.
Question 4.(10+16=26 points) Let G = (V1, E1) and H = (V2, E2) be the following graphs: a с u V b z W e d X G = (V1, E1) y H = (V2, E2) a) Draw the complement G of G. b) Show that G and H are isomorphic by writing a graph isomorphism F : V1 + V2.
19. Use the definition below and the minimum criteria a graph must meet in order to be potentially isomorphic to answer the question Recall Definition: Isomorphism: For a graph G1 V, E13 and G2-V2, E23 G1is isomorphic to G2 denoted GG2 iff af:V V2where i) f is bijective and Describe an isomorphism between the following two graphs, or briefly explain why no such isomorphism exists. f(A)1 f(B)6 f(C) 3 f(D)8 IG)2 f(H)5 f(I)4
19. Use the definition below and the...
Graphs (15 points) 14. For the following graph (8 points): a. Find all the edges that are incident of v1: b. Find all the vertices that are adjacent to v3: C. Find all the edges that are adjacent to e1: d. Find all the loops: e. Find all the parallel edges: f. Find all the isolated vertices: g. Find the degree of v3: h. Find the total degree of the graph: e3 e2 V2 VI 26 e4 e7 es 05...
Lemma. If two vector spaces have the same dimension then they are isomorphic Proof. To show that any two spaces of dimension n are isomorphic, we can simply show that any one is isomorphic to R. Then we will have shown that they are isomorphic to each other, by the transitivity of isomorphism (which was established in the first Theorem of this section) Theorem 1 Isomorphism is an equivalence relation among ctor spaces Let v be n--dimensional. Fix a basis...
Please answer part C in detail:
Problem 1: The following two NFAs, G1 and G2, represent behaviors of a certain discrete-event system. In this question, you are expected to show steps of your construction method. You may use software to help you with your construction but a single screenshot of the end result, without any explanation or construction steps, will not be accepted. G1: Marks sequences that end with a suffix bbc G2: Marks sequences that end with a suffix...
Problem 5. (12 marks) Connectivity in undirected graphs vs. directed graphs. a. (8 marks) Prove that in any connected undirected graph G- (V, E) with VI > 2, there are at least two vertices u, u є V whose removal (along with all the edges that touch them) leaves G still connected. Propose an efficient algorithm to find two such vertices. (Hint: The algorithm should be based on the proof or the proof should be based on the algorithm.) b....
Please do only e and f and show work
null(AT) null(A) T col(A) row(A) Figure 5.6 The four fundamental subspaces (f) Find bases for the four fundamental subspaces of 1 1 1 6 -1 0 1 -1 2 A= -2 3 1 -2 1 4 1 6 1 3 8. Given a subspace W of R", define the orthogonal complement of W to be W vE R u v 0 for every u E W (a) Let W span(e, e2)...
Most questions answered within 3 hours.
-
Discuss the sales orders process module(s) in the Enterprise
Resource Planning system. How does it contribute...
asked 15 minutes ago -
Which of the eight traits/ skills associated with being an
effective project manager is the most...
asked 22 seconds from now -
In the Metric System, the basic unit of measurement for length
is ___
In the Metric...
asked 6 minutes ago -
A spatially uniform electric field varies in time according
to E = Eo + 3000 t,...
asked 7 minutes ago -
A sample of C3H8 has 1.60×1024 H atoms.
How many carbon atoms does the sample contain?...
asked 13 minutes ago -
The position in an object as a function of time is given as ?
(?) =...
asked 16 minutes ago -
Design a class for python named PersonData with the following
member variables:
lastName
firstName
address
city...
asked 29 minutes ago -
How does use of the risk/need/responsivity model impact
effective rehabilitation services?
asked 34 minutes ago -
Your rich uncle has just given you a high school graduation
present of $900,000. The present,...
asked 36 minutes ago -
1=Write a program in C to get 16-bit data from Port-D and send
it to ports...
asked 35 minutes ago -
Calculate Ecell for the following reaction and conditions: 0.50
M Br2 (aq), 0.10 M Pb+2 (aq),...
asked 55 minutes ago -
There can be more than one correct answer.
Hypophysiotropic hormones:
A. released by the hypothalamus
B....
asked 1 hour ago