**Graph Theory Quiz Question:**Question: An undirected graph \( V \) consists of vertices \( A, B, C, D, E, F \) and edges \( (A,B), (A,C), (A,D), (B,C), (B,D), \) and \( (C,D) \). Graph \( V \) is:Options:- O None of these is correct- O a tree- O complete- O disconnected- O fullConsider the structure and properties of graph \( V \) when selecting from the given options. Analyze which characteristics apply to determine the correct classification of the graph.

Answered: Undirected graph V consists of vertices…

Undirected graph V consists of vertices A, B, C, D, E, F and edges (A,B), (A,C), (A,D), (B,C), (B,D) and (C,D Graph V is: O full. O disconnected. O complete O a tree O None of these is correct

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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Depth First Search

Depth-first search (DFS) also known as Depth First traversal is an algorithm used to traverse or search the nodes in a tree or graph data structure. Traversal usually means visiting all the nodes of a graph. DFS performs in a depth-ward manner. It uses a stack to remember the vertices it has visited. The search starts from the root node, where any arbitrary node is selected as a root node in a graph, and then each node is explored along each branch before performing backtracking. The DFS algorithm uses a recursive technique by following the idea of backtracking. DFS algorithm does this by performing an in-depth search of all the nodes by traversing in a forward direction. If the required node is not found, it starts backtracking to previously visited nodes.

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Recursively, a complete binary tree is a graph. Basis step: One vertex forms a binary tree. Inductive step: If T1 and T2 are disjoint complete binary trees with roots r1, r2, respectively, the graph produced by starting with a root r and adding an edge from r to each of the vertices r1, r2 is also complete. Recursively define a complete binary tree's leaves. Base step: The full binary tree has one root, r. Inductive step: T's leaves are the union of T1's and T2's. Class defines binary tree height h(T). Use structural induction to prove that `(T), the number of leaves of a complete binary tree T, is less than 2 h(T).
A complete binary tree is a graph defined through the following recur- sive definition. Basis step: A single vertex is a complete binary tree.Inductive step: If T1 and T2 are disjoint complete binary trees with roots r1, r2, respectively, the the graph formed by starting with a root r, and adding an edge from r to each of the vertices r1,r2 is also a complete binary tree.The set of leaves of a complete binary tree can also be defined recur- sively.Basis step: The root r is a leaf of the complete binary tree with exactly one vertex r.Inductive step: The set of leaves of the tree T built from trees T1, T2 is the union of the sets of leaves of T1 and the set of leaves of T2. The height h(T ) of a binary tree is defined in the class.Use structural induction to show that L(T), the number of leaves of a complete binary tree T , satisfies the following inequality L(T) ≤ 2^h(T).
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Write Kruskal’s algorithm with the union–find data structure incorporated is as follows. pre-cond: G is an undirected graph. post-cond: The output consists of a minimal spanning tree.
Consider the following graph. If we run breadth first search on this graph starting at any vertex, which one of the following is a possible order for visiting the nodes? NQMPOR QMNPOR MNOPQR QMNPRO Assume the following adjacency list for a graph. After you draw the graph, what is the path to reach D from A. Vertex Edges A --------> B, C B --------> C C --------> B, E D --------> B E --------> B, D Both (B) and (C) A->B->C->E->D A->B->C->D A->C->E->D
Consider the following graph. Start from vertex x, use the Depth First Search (DFS) algorithm (described in the textbook) to compute the DFS tree(s). Show intermediate results similar to the textbook example and show the final DFS tree(s). It is assumed that in the adjacency list representation, all vertices are stored according to their alphabetical order. In other words, a vertex’s neighbors should be explored by their relative alphabetical order.
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