Give a definition of Graph Embedding, Planar Embedding. Is the following graph a planar graph? G={V,E} V={a,b,c,d} E={{a,b},{b,c},{c,d},{a,d},{a,c},{d,b}} Show how BFS and DFS will work on the following directed graph and the order in which nodes will be discovered starting with Node A. G=(V,E); V={A,B,C,D,F}; E={(A,B), (A,C), (A,D), (C,D), (B,F),(D,F)} What are 2 main differences between Free Tree and Rooted Tree?

Give a definition of Graph Embedding, Planar Embedding. Is the following graph a planar graph? G={V,E} V={a,b,c,d} E={{a,b},{b,c},{c,d},{a,d},{a,c},{d,b}} Show how BFS and DFS will work on the following directed graph and the order in which nodes will be discovered starting with Node A. G=(V,E); V={A,B,C,D,F}; E={(A,B), (A,C), (A,D), (C,D), (B,F),(D,F)} What are 2 main differences between Free Tree and Rooted Tree?

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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In this task you will work with an undirected Graph G = {V, E}, where V = {f,p,s,b,l,j,t,c,d} and E = {(1,2), (1,3), (3,8), (4,8), (8,9), (1,7), (2,6), (2,3), (5,6), (6,7), (7,9), (8,1)}. Assume that the nodes are stored in an indexed linear structure (e.g., an array or a vector) numbered consecutively from 1 (node f) to 9 (node d). Run, by hand, a depth-first search (DFS) traversal of the graph G, starting at node p. When choosing which node to visit next amongst the possibilities, choose the one that is next in alphabetical order. Give your answer by listing the edges in the order that DFS will select.
In this task you will work with an undirected Graph G = {V, E}, where V = {f,p,s,b,l,j,t,c,d} and E = {(1,2), (1,3), (3,8), (4,8), (8,9), (1,7), (2,6), (2,3), (5,6), (6,7), (7,9), (8,1)}. Assume that the nodes are stored in an indexed linear structure (e.g., an array or a vector) numbered consecutively from 1 (node f) to 9 (node d). 1. Type up a tgf representation of the graph 2. Find a path from node l to node b with length 8, that passes through every vertex of the graph. List the nodes of that path. 3. The graph G contains cycles. What is the smallest number of vertices to remove in order to break all cycles?
A directed graph G= (V,E) consists of a set of vertices V, and a set of edges E such that each element e in E is an ordered pair (u,v), denoting an edge directed from u to v. In a directed graph, a directed cycle of length three is a triple of vertices (x,y,z) such that each of (x,y) (y,z) and (z,x) is an edge in E. Write a Mapreduce algorithm whose input is a directed graph presented as a list of edges (on a file in HDFS), and whose output is the list of all directed cycles of length three in G. Write the pseudocode for the mappers/reducers methods. Also, assuming that there are M mappers, R reducers, m edges and n vertices -- analyze the (upper-bound of the) communication cost(s).
Let G = (V, E) be a graph with vertex-set V = {1, 2, 3, 4, 5} and edge-set E = {(1, 2), (3, 2), (4, 3), (1, 4), (2, 4), (1, 3)}. (a) Draw the graph. Find (b) maximal degree, i.e. ∆(G), (c) minimal degree, i.e. δ(G), (d) the size of biggest clique, i.e. ω(G),(e) the size of biggest independent set, i.e. α(G), ter(f) the minimal number of colours needed to color the graph, i.e. χ(G).
let a graph have vertices h,i,j,k,l,m,n,o and edge set {{h,i},{h,j},{h,k},{h,n},{h,o},{j,k},{j,l},{j,m},{k,l},{m,o},. on paper, draw the graph. then answer the questions below. a. what is the degree of vertex k? b. what is the degree of vertex h? c. how many connected components does the graph have?
Consider a set of real number pairs that represent a collection of V intervals on the real line. An interval graph with one vertex for each interval is defined by such a collection, with edges connecting vertices in the event that the associated intervals cross (have any points in common). Create a programme that creates V random intervals, each of length d, in the unit interval and then creates the interval graph that goes with it.Use a BST, as a tip.
Suppose that G is an unconnected graph that consists of 4 connected components. The first component is K4, the second is K2,2, the third is C4 and the fourth is a single vertex. Your job is to show how to add edges to G so that the graph has an Euler tour. Justify that your solution is the minimum number of edges added.
Consider the sets of A₁, A₂, ..., Aₙ. The intersection graph of a collection of sets A₁, A₂, ..., Aₙ is the graph that has a vertex for each of these sets and has an edge connecting the vertices representing two sets if these sets have a nonempty intersection. Construct the intersection graph for the sets given below. A₁ = {0, 2, 4, 6, 8), A₂ = {0, 1, 2, 3, 4), A₃ = {1,3,5,7,9), A₄ = {5, 6, 7, 8, 9), A₅ = {0, 1,8,9} Note: A₁ {0, 2, 4, 6,8} represents vertex A₁ (first vertex). This set defines the possible edge connections of the corresponding vertex.
For the following simple graphs G_1=(V_1,E_1)G1=(V1,E1) and G_2=(V_2,E_2)G2=(V2,E2) (described by their vertex and edge sets) decide whether they are isomorphic. If they are then prove it, and if they are not give a convincing argument that explains why not: V_1=\{ a,b,c,d,e,f,g,h\}V1={a,b,c,d,e,f,g,h}, E_1=\{ ac,ag,ah,bc,bd,bh,ce,de,df,fg,fh\}E1={ac,ag,ah,bc,bd,bh,ce,de,df,fg,fh}, V_2=\{ 1,2,3,4,5,6,7,8\}V2={1,2,3,4,5,6,7,8}, E_2=\{ 12,15,16,25,27,34,36,37,38,48,67\}E2={12,15,16,25,27,34,36,37,38,48,67}
Throughout, a graph is given as input as an adjacency list. That is, G is a dictionary where the keysare vertices, and for a vertex v,G[v] = [u such that there is an edge going from v to u].In the case that G is undirected, for every edge u − v, v is in G[u] and u is in G[v]. 3. Write the full pseudocode for the following problem.Input: A directed graph G, and an ordering on the vertices given in a list A.Output: Is A a topological order? In other words, is there an i, j such that i < j and there is an edge fromA[j] to A[i]?
We are given an undirected connected graph G = (V, E) and vertices s and t.Initially, there is a robot at position s and we want to move this robot to position t by moving it along theedges of the graph; at any time step, we can move the robot to one of the neighboring vertices and the robotwill reach that vertex in the next time step.However, we have a problem: at every time step, a subset of vertices of this graph undergo maintenance andif the robot is on one of these vertices at this time step, it will be destroyed (!). Luckily, we are given theschedule of the maintenance for the next T time steps in an array M [1 : T ], where each M [i] is a linked-listof the vertices that undergo maintenance at time step i.Design an algorithm that finds a route for the robot to go from s to t in at most T seconds so that at notime i, the robot is on one of the maintained vertices, or output that this is not possible. The runtime ofyour algorithm should ideally be O((n + m) ·T ) but you will…
Computer Science Frequently, a planar graph G=(V,E) is represented in the edgelist form, which for each vertex vi V contains the list of its incident edges, arranged in the order in which they appear as one proceeds counterclockwise around i v . Show that the edge-list representation of G can be transformed to the DCEL (DoublyConnected-Edge-List) representation in time O(|V|).