Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 34.2, Problem 1E
Program Plan Intro
To prove that GRAPH-ISOMORPHISM belongs to NP by illustrating a polynomial-time
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Recall the Clique problem: given a graph G and a value k, check whether G has a set S of k vertices that's a clique. A clique is a subset of vertices S such that for all u, v € S, uv is an edge of G. The goal of this problem is to establish the NP-hardness of Clique by reducing VertexCover, which is itself an NP-hard problem, to Clique.
Recall that a vertex cover is a set of vertices S such that every edge uv has at least one endpoint (u or v) in S, and the VertexCover problem is given a graph H and a value 1, check whether H has a vertex cover of size at most 1. Note that all these problems are already phrased as decision problems, and you only need to show the NP-Hardness of Clique. In other words, we will only solve the reduction part in this problem, and you DO NOT need to show that Clique is in NP.
Q4.1
Let S be a subset of vertices in G, and let C be the complement graph of G (where uv is an edge in C if and only if uv is not an edge in G).
Prove that for any subset of vertices…
Please Answer this in Python language:
You're given a simple undirected graph G with N vertices and M edges. You have to assign, to each
vertex i, a number C; such that 1 ≤ C; ≤ N and Vi‡j, C; ‡ Cj.
For any such assignment, we define D; to be the number of neighbours j of i such that C; < C₁.
You want to minimise maai[1..N) Di - mini[1..N) Di.
Output the minimum possible value of this expression for a valid assignment as described above,
and also print the corresponding assignment.
Note:
The given graph need not be connected.
• If there are multiple possible assignments, output anyone.
• Since the input is large, prefer using fast input-output methods.
Input
1
57
12
13
14
23
24
25
35
Output
2
43251
Q
Code in Python Please
For an undirected graph G with n vertices, prove the following are equivalent:
G is a tree
G is connected, but if any edge is removed the resulting graph is not connected.
For any two distinct vertices , u and v , there is exactly one simple path from u to v.
G contains no cycles and has n-1 edges
Chapter 34 Solutions
Introduction to Algorithms
Ch. 34.1 - Prob. 1ECh. 34.1 - Prob. 2ECh. 34.1 - Prob. 3ECh. 34.1 - Prob. 4ECh. 34.1 - Prob. 5ECh. 34.1 - Prob. 6ECh. 34.2 - Prob. 1ECh. 34.2 - Prob. 2ECh. 34.2 - Prob. 3ECh. 34.2 - Prob. 4E
Ch. 34.2 - Prob. 5ECh. 34.2 - Prob. 6ECh. 34.2 - Prob. 7ECh. 34.2 - Prob. 8ECh. 34.2 - Prob. 9ECh. 34.2 - Prob. 10ECh. 34.2 - Prob. 11ECh. 34.3 - Prob. 1ECh. 34.3 - Prob. 2ECh. 34.3 - Prob. 3ECh. 34.3 - Prob. 4ECh. 34.3 - Prob. 5ECh. 34.3 - Prob. 6ECh. 34.3 - Prob. 7ECh. 34.3 - Prob. 8ECh. 34.4 - Prob. 1ECh. 34.4 - Prob. 2ECh. 34.4 - Prob. 3ECh. 34.4 - Prob. 4ECh. 34.4 - Prob. 5ECh. 34.4 - Prob. 6ECh. 34.4 - Prob. 7ECh. 34.5 - Prob. 1ECh. 34.5 - Prob. 2ECh. 34.5 - Prob. 3ECh. 34.5 - Prob. 4ECh. 34.5 - Prob. 5ECh. 34.5 - Prob. 6ECh. 34.5 - Prob. 7ECh. 34.5 - Prob. 8ECh. 34 - Prob. 1PCh. 34 - Prob. 2PCh. 34 - Prob. 3PCh. 34 - Prob. 4P
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- Graphs G and H are isomorphic if the nodes of G may be reordered so that it is identical to H. Let ISO = { | G and H are isomorphic graphs}. Show that ISO is in the class NP.arrow_forwardThe subgraph-isomorphism problem takes two graphs G1 and G2 and asks whether G1 is isomorphic to a subgraph of G2. Show that a) the subgraph-isomorphism problem is in NP; b) it is NP-complete by giving a polynomial time reduction from SAT problem to it. and Note: Two graphs G1=(V1, E1) and G2=(V2, E2) are isomorphic if there exists a one-one and onto function f() from V1 to V2 such that for every two nodes u and v in V1, (u.v) is in E1 if and only if (f(u), f(v)) is in E2. For examples, G1 is isomorphic to a subgraph with vertices {1,2,5,4} of G2 below. G1 G2 1 1 2 4 5 4 3. 2. 3.arrow_forwardFor each pair of graphs G1 = <V1, E1> and G2 = <V2, E2> a) determine if they are isomorphic or not. b) Determine a function that can be isomorphic between them if they are isomorphic. Otherwise you should justify why they are not isomorphic. c) is there an Euler road or an Euler bike in anyone graph? Is Hamilton available? You should draw if the answer is yes and reason if your answer is no.arrow_forward
- 1. Let G be a tree and let L be the set of leaves in G (the vertices of degree 1). (a) If G contains a vertex of degree k, show that |L| > k. (b) Let f be a graph isomorphism from G to G. Prove that f(L) = L. (c) Prove that either there is a vertex v e V(G) such that f(v) = v or there is an edge {x, y} € E(G) such that {f(x), f(y)} = {x, y}. (Hint: Induction on |V(G)|; in the induction step consider a restriction of ƒ to a subset of vertices.)arrow_forwardMake java program that gives results to determine if a connected simple graph has independent sets & identify the minimum set & maximum independent set. Output of the program: - independent sets - maximum independent set - minimum independent setarrow_forwardnot handwritten Find with proof every graph G for which ex(n, G) is defined for all positive integers n andthere exists a constant c such that ex(n, G) = ex(m, G) for all integers m, n > c.arrow_forward
- The third-clique problem is about deciding whether a given graph G = (V, E) has a clique of cardinality at least |V |/3.Show that this problem is NP-complete.arrow_forwardQuestion 1 Let S be a subset of vertices in G, and let C be the complement graph of G (where uv is an edge in C if and only if uv is not an edge in G).Prove that for any subset of vertices S, S is a vertex cover in G if and only if V\S is a clique in C.Note: this is an if and only if proof, i.e. you need to show both directions for full credit. Question 2 Part 4.1 implies the following result (which you may use without proof): G has a vertex cover of size at most k if and only if the complement of G has a clique of size at least n−k.Use this fact to give a reduction from VertexCover to Clique. Your solution should have the following two steps:i) First, show the reduction: specify how the inputs to VertexCover, G and k, can be transformed to a valid input pair, H and l, for Clique. Make sure to explain why this takes polynomial time.ii) Second, show that the answer to Clique(H,l) can be converted to the answer of VertexCover(G,k). One possibility is to explain how a YES answer to…arrow_forwardExercise c. Strict matchingConsider the following generalization of the maximum matching problem, which we callStrict-Matching. Recall that a matching in an undirected graph G = (V, E) is a setof edges such that no distinct pair of edges {a, b} and {c, d} have endpoints that areequal: {a, b} ∩ {c, d} = ∅. Say that a strict matching is matching with the propertythat no pair of distinct edges have endpoints that are connected by an edge: {a, c} ̸∈ E,{a, d} ̸∈ E, {b, c} ̸∈ E, and {b, d} ̸∈ E. (Since a strict matching is also a matching, wealso require {a, b} ∩ {c, d} = ∅.) The problem Strict-Matching is then given a graphG and an integer k, does G contain a strict matching with at least k edges.Prove that Strict-Matching is NP-complete.arrow_forward
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