given a graph G, and a subset S of its vertices, find a spanning tree of G such that the set S is exactly the set of leaves of that spanning tree. Prove or disprove that this problem is NP-complete.
Let us consider the following Graph G

Kruskal’s Algorithm to find the Minimum Cost Spanning Tree (MCST) of a graph G as follows:
Step 1:
Step 2:
Step 3:

Step 4:
Step 5:
Step 6:

For n- Vertices we get (n-1) Edges in MCST. Here in graph having 7 Vertices so we get 6 Edges
Which is Required MCST of a graph
given a graph G, and a subset S of its vertices, find a spanning tree of...
Let G = (V;E) be an undirected and unweighted graph. Let S be a subset of the vertices. The graph induced on S, denoted G[S] is a graph that has vertex set S and an edge between two vertices u, v that is an element of S provided that {u,v} is an edge of G. A subset K of V is called a killer set of G if the deletion of K kills all the edges of G, that is...
7. An independent set in a graph G is a subset S C V(G) of vertices of G which are pairwise non-adjacent (i.e., such that there are no edges between any of the vertices in S). Let Q(G) denote the size of the largest independent set in G. Prove that for a graph G with n vertices, GX(G)n- a(G)+ 1.
Let G be an undirected graph and let X be a subset of the vertices of G. A connecting tree on X is a tree composed out of the edges of G that contains all the vertices in X. One way to compute a connecting tree consists of two steps: (1) Compute a minimum spanning tree T over G. (2) Delete all the edges out of T not needed to connect vertices in X. Give an algorithm(Pseudo-code) to carry out...
Let G be an undirected graph and let X be a subset of the vertices of G. A connecting tree on X is a tree composed out of the edges of G that contains all the vertices in X. One way to compute a connecting tree consists of two steps: (1) Compute a minimum spanning tree T over G. (2) Delete all the edges out of T not needed to connect vertices in X. The Steiner tree for X is...
Problem 3: Bounded-Degree Spanning Trees (10 points). Recall the minimum spanning tree problem studied in class. We define a variant of the problem in which we are no longer concerned with the total cost of the spanning tree, but rather with the maximum degree of any vertex in the tree. Formally, given an undirected graph G = (V,E) and T ⊆ E, we say T is a k-degree spanning tree of G if T is a spanning tree of G,...
You are given an undirected graph G with weighted edges and a minimum spanning tree T of G. Design an algorithm to update the minimum spanning tree when the weight of a single edge is increased. The input to your algorithm should be the edge e and its new weight: your algorithm should modify T so that it is still a MST. Analyze the running time of your algorithm and prove its correctness.
You are given an undirected graph G with weighted edges and a minimum spanning tree T of G. Design an algorithm to update the minimum spanning tree when the weight of a single edge is decreased. The input to your algorithm should be the edge e and its new weight; your algorithm should modify T so that it is still a MST. Analyze the running time of your algorithm and prove its correctness.
The input to SPANNINGTREEWITHKLEAVES is a graph G and an integer K. The question asked by SPAN NINGTREEWITHKLEAVES is whether G has a spanning tree with exactly K leaves. Problem 3. Show that SPANNINGTREEWITIIKLEAVES is NP-complete. Hint: There is a simple polynomial time reduction from HAMILTONIANPATH to SPANNINGTREEWITHKLEAVES.
2. This question concerns the graph G shown below. (a) Mark the spanning tree for G obtained by performing a depth-first search starting at the vertex A, and using the convention that nearby vertices should be explored in a counter-clockwise fasion, beginning with east; so E comes first, then NE, then N, ... (b) Mark the spanning tree for G obtained by performing a breadth-first search starting at the vertex A, and using the convention that nearby vertices should be...
2. This question concerns the graph G shown below. (a) Mark the spanning tree for G obtained by performing a depth-first search starting at the vertex A, and using the convention that nearby vertices should be explored in a counter-clockwise fasion, beginning with east; so E comes first, then NE, then N, ... (b) Mark the spanning tree for G obtained by performing a breadth-first search starting at the vertex A, and using the convention that nearby vertices should be...