The subgraph-isomorphism problem takes two graphs G1 and G2 and askswhether G1 is isomorphic to a subgraph of G2. Show thata) the subgraph-isomorphism problem is in NP;b) it is NP-complete by giving a polynomial time reduction from SAT problem to it.andNote: Two graphs G1=(V1, E1) and G2=(V2, E2) are isomorphic if there exists aone-one and onto function f() from V1 to V2 such that for every two nodes u andv in V1, (u.v) is in E1 if and only if (f(u), f(v)) is in E2. For examples, G1 isisomorphic to a subgraph with vertices {1,2,5,4} of G2 below.G1G21124543.2.3.

The subgraph-isomorphism problem takes two graphs G1 and G2 and asks vhether G1 is isomorphic to a subgraph of G2. Show that a) the subgraph-isomorphism problem is in NP; and ) it is NP-complete by giving a polynomial time reduction from SAT problem to it. 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 in V1, (u.v) is in E1 if and only if (f(u), f(v)) is in E2. For examples, G1 is somorphic to a subgraph with vertices {1,2,5,4} of G2 below. G1 G2 1 1 2 3 4 5 3

The subgraph-isomorphism problem takes two graphs G1 and G2 and asks vhether G1 is isomorphic to a subgraph of G2. Show that a) the subgraph-isomorphism problem is in NP; and ) it is NP-complete by giving a polynomial time reduction from SAT problem to it. 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 in V1, (u.v) is in E1 if and only if (f(u), f(v)) is in E2. For examples, G1 is somorphic to a subgraph with vertices {1,2,5,4} of G2 below. G1 G2 1 1 2 3 4 5 3

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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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…
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The problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy. ALso provide an expression using qualifiers. YOU ALSO NEED TO PROVIDE A CLEAR EXPRESSION USING THE QUALIFIERS.PLEASE GIVE CLEAR EXPLAINATION.
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The problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy. ALso provide an expression using qualifiers. You need to provide a clear expression using the qualifiers.Please give clear explaination
The problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy.
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Let G = (V, E) be a connected, undirected graph, and let s be a fixed vertex in G. Let TB be the spanning tree of G found by a bread first search starting from s, and similarly TD the spanning tree found by depth first search, also starting at s. (As in problem 1, these trees are just sets of edges; the order in which they were traversed by the respective algorithms is irrelevant.) Using facts, prove that TB = TD if and only if TB = TD = E, i.e., G is itself a tree.