consider the following problem: Given a Graph G = (V, E), does G have a cycle? Show that this problem is in NP.
consider the following problem: Given a Graph G = (V, E), does G have a cycle? Show that this problem is in NP.
Prove that the following problem is NP-complete: given an undirected graph G = (V, E) and an integer k, return a clique of size k as well as an independent set of size k, provided both exist.
9. Consider the graph in problem 8, call it G. a) Find at least one non-trivial graph automorphism on G. That is, find a graph isomorphism f:G -G. Show that there are bijective mappings g: V(G)-V(G) and h: E(G)-E(G). Show that the mappings preserve the edge-endpoint function for G. b) Find a mapping fl:G G that is the inverse of the automorphism you found in part a c) Show that fof- I, which is the identity automorphism that sends each...
Write the definition of G. Does the graph has a Hamiltonian cycle? If yes, show it, if not why ? Does the graph have a Euler cycle? If yes, show it, if not why ? Is this graph bipartite? If yes show your partitions Consider the following graph G Write the definition of G
3. Given graph G = (V,E), prove that the following statements are equivalent. [Note: the following statements are equivalent definitions of a "tree graph".] 1) There exist exactly one path between any of two vertices u, v EV in the graph G 2) Graph G is connected and does not contain any cycles. 3) Graph G does not contain any cycles, and a cycle is formed if any edge (u, v) E E is added to G 3. Given graph...
3, (30 points) Given a directed graph G - N. E), each edge eEhas weight We, 3, (30 points) Given a directed graph G (V, E), each edgee which can be positive or negative. The zero weight cycle problem is that whether exists a simple cycle (each vertex passes at most once) to make the sum of the weights of each edge in G is exactly equal to 0. Prove that the problem is NP complete. 3, (30 points) Given...
(a) Given a graph G = (V, E) and a number k (1 ≤ k ≤ n), the CLIQUE problem asks us whether there is a set of k vertices in G that are all connected to one another. That is, each vertex in the ”clique” is connected to the other k − 1 vertices in the clique; this set of vertices is referred to as a ”k-clique.” Show that this problem is in class NP (verifiable in polynomial time)...
Show that the following problem is NP-Complete (Hint: reduce from 3-SAT or Vertex Cover). Given an undirected graph G with positive integer distances on the edges, and two integers f and d, is there a way to select f vertices on G on which to locate firehouses, so that no vertex of G is at distance more than d from a firehouse?
Updating an MST when an edge weight changes. You have a graph G= (V, E) with edge weights given in the graph (whatever they are). In addition, a minimum spanning tree T= (V, E′) of this graph has also been given to you. Now, say we need to increase the weight of one particular edge e. Does the MST change? If so, show how to compute the new MST in linear time. You should consider two cases: 1). when e∈E′and...
1) Consider the clique problem: given a graph G (V, E) and a positive integer k, determine whether the graph contains a clique of size k, i.e., a set of k vertices S of V such that each pair of vertices of S are neighbours to each other. Design an exhaustive-search algorithm for this problem. Compute also the time complexity of your algorithm.
The Max Cut problem is given a undirected graph G(V, E), finding a set S so that the number of edges that go between S and V − S is maximum. This is an NPC problem. a) Show that there is always a max cut of size at least |E|/2. Hint: Decide where to put vertices according to if they have more neighbors in S or V − S.