Given a directed graph, prove that there exists an Eulerian cycle that is also a hamiltonian...

Given a directed graph, prove that there exists an Eulerian cycle that is also a hamiltonian cycle if and only if the graph is a single cycle.

Expert Solution

Colby Messinger answered 2 years ago

You are given an undirected graph A. The problem P asks, if there exists a cycle...

You are given an undirected graph A. The problem P asks, if there exists a cycle that contains more than half of the nodes in graph A? Is this problem in NP(nondeterministic polynomial time)? How is it related to the Hamiltonian cycle problem? Construct your example in python. Input: Graph A as either an edge list or adjacent list. Output: Yes or No. If the output is yes, display the cycle as a sequence of nodes.

Prove a connected simple graph G with 16 vertices and 117 edges is not Eulerian.

Let G be a graph. prove G has a Eulerian trail if and only if G...

Let G be a graph. prove G has a Eulerian trail if and only if G has at most one non-trivial component and at most 2 vertices have odd degree

Assume the input graph is directed but may or may not have cycle in it, that...

Assume the input graph is directed but may or may not have cycle in it, that is, may or may not be a directed acyclic graph. Consider the recursive topological sorting algorithm shown below. TopologicalSort(G) { If G has only one node v then return v. Find a node v with no incoming edge and remove v from G. Return v followed by the order returned by TopologicalSort(G). } (a) Extend the recursive algorithm shown below so it can detect...

Prove that \strongly connected" is an equivalence relation on the vertex set of a directed graph

) Eulerian and Hamiltonian cycles. Consider the graphs defined by the following four sets of edges:...

) Eulerian and Hamiltonian cycles. Consider the graphs defined by the following four sets of edges: a) 0-1 0-2 0-3 1-3 1-4 2-5 2-9 3-6 4-7 4-8 5-8 5-9 6-7 6-9 7-8 b) 0-1 0-2 0-3 1-3 0-3 2-5 5-6 3-6 4-7 4-8 5-8 5-9 6-7 6-9 8-8 c) 0-1 1-2 1-3 0-3 0-4 2-5 2-9 3-6 4-7 4-8 5-8 5-9 6-7 6-9 7-8 d) 4-1 7-9 6-2 7-3 5-0 0-2 0-8 1-6 3-9 6-3 2-8 1-5 9-8 4-5 4-7...

Java: Determining whether a tree exists in a directed graph I'm trying to figure out how...

Java: Determining whether a tree exists in a directed graph I'm trying to figure out how to determine if a tree exists in a directed graph, I need help with the isTree() function. the code for the Bag class used is below the main code if it's needed. import algs13.Bag; import java.util.HashSet; // See instructions below public class MyDigraph { static class Node { private String key; private Bag<Node> adj; public Node (String key) { this.key = key; this.adj =...

An eulerian walk is a sequence of vertices in a graph such that every edge is...

An eulerian walk is a sequence of vertices in a graph such that every edge is traversed exactly once. It differs from an eulerian circuit in that the starting and ending vertex don’t have to be the same. Prove that if a graph is connected and has at most two vertices of odd degree, then it has an eulerian walk.

Draw a graph having the given properties stated below, or explain why no such graph exists:...

Draw a graph having the given properties stated below, or explain why no such graph exists: In each case assume simple graphs (no self loops and no parallel edges) a. Six vertices each with degree 3 b. Five vertices each with degree 3. c. Four vertices each with degree 1. d. Six vertices and four edges. e. Four edges; four vertices having degrees 1, 2, 3, and 4.

For each of the following sets, prove that thay are convex sets or not. Also graph...

For each of the following sets, prove that thay are convex sets or not. Also graph the sets. a) ? 1= {(?1 , ?2 ): ?1 ^2 + ?2^2 ≥ 1} b)?2 = {(?1 ,?2 ): ?1 ^2 + ?2^ 2 = 1} c)?3 = {(?1 , ?2 ): ?1 ^2 + ?2 ^2 ≥ 1}

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