Introduction to Algorithms
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 21.1, Problem 3E
Program Plan Intro

To find the number of times FIND-SET and UNION functions are called during the execution of CONNECTED-COMPONENETS on an undirected graph G=(V,E) .

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An Euler path, in a graph or multigraph, is a walk through the graph which uses every edge exactly once. An Euler circuit is an Euler path which starts and stops at the same vertex.  NOTE: graphs are in the image attached. Which of the graphs below have Euler paths? Which have Euler circuits? List the degrees of each vertex of the graphs above. Is there a connection between degrees and the existence of Euler paths and circuits? Is it possible for a graph with a degree 1 vertex to have an Euler circuit? If so, draw one. If not, explain why not. What about an Euler path? What if every vertex of the graph has degree 2. Is there an Euler path? An Euler circuit? Draw some graphs. Below is part of a graph. Even though you can only see some of the vertices, can you deduce whether the graph will have an Euler path or circuit? NOTE: graphs is in the image attached.
Be G=(V, E)a connected graph and u, vEV. The distance Come in u and v, denoted by d(u, v), is the length of the shortest path between u'and v, Meanwhile he width from G, denoted as A(G), is the greatest distance between two of its vertices. a) Show that if A(G) 24 then A(G) <2. b) Show that if G has a cut vertex and A(G) = 2, then Ġhas a vertex with no neighbors.
Given an undirected graph G, and a path P, we want to verify that P is a cycle that contains all nodes in the graph (each node occurs on P exactly once, no repetition is allowed).In numbered steps, describe a polynomial-time verifier that checks if a given path P has the described properties (meaning that it is a cycle and it contains all nodes in the graph, with each node appearing on the path exactly once).(Hint: you can think of it in terms of this question: what does the verifier need to check to ensure that the path P has the described properties?).
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