Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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G = (V,E,W) is a weighted connected (undirected) graph where all edges have distinct weights except two edges e and e′ which have the same weight. Suppose there is a Minimum Spanning Tree of G containing both e and e′. Prove that G has a unique Minimum Spanning Tree.
Let G be a directed acyclic graph with exactly one source r such that for any other vertex v there exists a unique directed path from r to v. Let Gu be the undirected graph obtained by erasing the direction on each edge of G. Prove that (Gu,r) is a rooted tree.
Show that the smallest edge must be present in a graph component's minimal spanning tree.
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- Let e be a maximum-weight edge on some cycle of graph G=(V, E). Prove or disprove that after deleting edge a from graph, there is a minimum spanning tree of G=(V, E-{e}) that is also a minimum spanning tree of G. Step by step answer.arrow_forwardSuppose you are given a connected graph G, with edge costs that are alldistinct. Prove that G has a unique minimum spanning tree.arrow_forwardSuppose G = (V, E) is an undirected connected weighted graph such that all its edge weightsare distinct. Prove that that the minimum spanning tree of G is unique.arrow_forward
- 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 for all u, v ∈ V , the path between u and v in tree T is uniquearrow_forwardLet G be a connected graph, and let T1, T2 be two spanning trees. Prove thatT1 can be transformed to T2 by a sequence of intermediate trees, each obtainedby deleting an edge from the previous tree and adding another.arrow_forwardLet the graph G be a cycle of n nodes in which x edges have the weight 100 and y edges have weight 200. How many minimum spanning trees does G have?arrow_forward
- Let G be a connected graph that has exactly 4 vertices of odd degree: v1,v2,v3 and v4. Show that there are paths with no repeated edges from v1 to v2, and from v3 to v4, such that every edge in G is in exactly one of these paths.arrow_forwardA (k, l)-dumbbell graph is obtained by taking a complete graph on k (labeled) nodes and a complete graph on l (labeled) nodes, and connecting them by a single edge. Find the number of spanning trees of a dumbbell graph.arrow_forwardProve that a minimum spanning tree of a graph component must include a shortest edgearrow_forward
- How can I prove that G/e is connected? (If G is a connected graph and e is an edge of G) And what does a spanning tree has to do with this proof?arrow_forwardAlgorithm :Let G be a connected graph and s a vertex of G. Thealgorithm determines the set C of cut points of G and the blocks of G and,thus, the number k of blocks of G).arrow_forwardLet G be a graph with a connected subgraph H. Prove that H is a subgraph of a unique connected component of G.arrow_forward
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