5. HC: Graph G has circuit that begins and ends at v1, visiting each vertex once.
HP: Graph G has a path that visits each vertex exactly once (begin anywhere, end anywhere)
a. Give yes and no instances of HC and HP
b. Prove or disprove: the following is a polynomial reduction from HC to HP
Given an undirected graph G=(V,E), create a new graph G’ by adding one new vertex s and |V| new edges, connecting s to every vertex in the original G.
5. HC: Graph G has circuit that begins and ends at v1, visiting each vertex once....
Note: For the following problems, you can assume that INDEPENDENT SET, VERTEX COVER, 3-SAT, HAMILTONIAN PATH, and GRAPH COLORING are NP-complete. You, of course, may look up the defini- tions of the above problems online. 5. The LONGEST PATH problem asks, given an undirected graph G (V, E), and a positive integer k , does G contain a simple path (a path visiting no vertex more than once) with k or more edges? Prove that LONGEST PATH is NP-complete. Note:...
Say that we have an undirected graph G(V, E) and a pair of vertices s, t and a vertex v that we call a a desired middle vertex . We wish to find out if there exists a simple path (every vertex appears at most once) from s to t that goes via v. Create a flow network by making v a source. Add a new vertex Z as a sink. Join s, t with two directed edges of capacity...
Hi,. the question is below: Help if you can.. Here is some background information/ an example: 9. Let k-Color be the following problem. Input: An undirected graph G. Question: Can the vertices of G be colored using k distinct colors, so that every pair of adjacent vertices are colored differently? Suppose that you were given a polynomial time algorithm for (k + 1)-Color. Use it to give a polynomial algorithm for k-Color. This means that you need to provide a...
Below is the Graph file that needs to be modified(using Python3) : #!/usr/bin/python3 # Simple Vertex class class Vertex: """ Lightweight vertex structure for a graph. Vertices can have the following labels: UNEXPLORED VISITED Assuming the element of a vertex is string type """ __slots__ = '_element', '_label' def __init__(self, element, label="UNEXPLORED"): """ Constructor. """ self._element = element self._label = label def element(self): """ Return element associated with this vertex. """ return self._element def getLabel(self): """ Get label assigned to...