5)
Let G = (V, E) be an undirected graph, with no parallel edges or self-loops. Let...
e hese seqd 7. Prove the handshaking theorem. Let G- (V,E) be an undirected graph with m edges. Then 2m Evev(degv)).
Remarks: All the graphs here are without self loops and parallel edges, and anti-parallel edges. When we speak of a flow network, we mean there are capacities c(e) ? 0 on the edges, the graph G is directed with a source s and a destination t. In all the algorithms, always explain their correctness and analyze their complexity. The complexity should be as small as possible. A correct algorithm with large complexity, may not get full credit. • Question 3:...
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...
Bounds on the number of edges in a graph. (a) Let G be an undirected graph with n vertices. Let Δ(G) be the maximum degree of any vertex in G, δ(G) be the minimum degree of any vertex in G, and m be the number of edges in G. Prove that δ(G)n2≤m≤Δ(G)n2
Let G = (V, E) be a weighted undirected connected graph that contains a cycle. Let k ∈ E be the edge with maximum weight among all edges in the cycle. Prove that G has a minimum spanning tree NOT including k.
2. Let G = (V, E) be an undirected connected graph with n vertices and with an edge-weight function w : E → Z. An edge (u, v) ∈ E is sparkling if it is contained in some minimum spanning tree (MST) of G. The computational problem is to return the set of all sparkling edges in E. Describe an efficient algorithm for this computational problem. You do not need to use pseudocode. What is the asymptotic time complexity of...
Let G= (V, E) be a connected undirected graph and let v be a vertex in G. Let T be the depth-first search tree of G starting from v, and let U be the breadth-first search tree of G starting from v. Prove that the height of T is at least as great as the height of U
Problem 3's picture are given below. 5. (a) Let G = (V, E) be a weighted connected undirected simple graph. For n 1, let cycles in G. Modify {e1, e2,.. . ,en} be a subset of edges (from E) that includes no Kruskal's algorithm in order to obtain a spanning tree of G that is minimal among all the spanning trees of G that include the edges e1, e2, . . . , Cn. (b) Apply your algorithm in (a)...
Problem 4 Let G = (V. E) be an undirected, connected graph with weight function w : E → R. Furthermore, suppose that E 2 |V and that all edge weights are distinct. Prove that the MST of G is unique (that is, that there is only one minimum spanning tree of G).
P9.6.3 Prove that a connected undirected graph G is bipartite if and only if there are no edges between nodes at the same level in any BFS tree for G. (An undirected graph is defined to be bipartite if its nodes can be divided into two sets X and Y such that all edges have one endpoint in X and the other in Y.) P9.6.3 Prove that a connected undirected graph G is bipartite if and only if there are...