Let G be a connected graph, and let T1, T2 be two spanning trees. Prove that T1 can be transformed to T2 by a sequence of intermediate trees, each obtained by deleting an edge from the previous tree and adding another.

Write Algorithm to computes SCDS of any vertex set R in a DP graph with essential dominating pair (u, v) with d(u, v) >9.

Let R be the reduction from 3-SAT to VERTEX-COVER as discussed in class.

Prove that a minimum spanning tree of a graph component must include a shortest edge. It is possible, in Dijkstra’s algorithm, that an edge removed from the priority queue is not useful: it takes us to a previously visited node. Some of these extraneous edges can be avoided by not placing an edge in the priority queue if the destination has already been visited. Is it still possible to encounter an edge to a previously visited node?

Let G = (V, E) denote an weighted undirected graph, in which every edge has unit weight, and let T = (V, E') denote the minimum spanning tree of G. Prove formally that the graph G need not have a unique tree T

Let Tsp be the tree of shortest paths of the below graph (G1) starting from vertex A and let TMST be a minimum spanning tree of (G1). Circle the only assertion that is FALSE 24 18. A 14 19 30 15 20 16 Both Tsp and TMST Contain the two edges of weight 6 Tsp and TMST have the same number of edges O Tsp and TMST have six edges in common The shortest path between A and H uses exactly 4 edges