5.40 Show for every connected graph G of diameter 2 or more and every two ver- tices u and v in G that G2 contains a proper u- v path but not necessarily two internally disjoint proper u -v paths...
Let G be a non-Hamiltonian, connected graph. For every pair of nonadjacent vertices u and v, 8(u) +8()2 k, for some k> O. Show that G contains a path of length k. Let G be a non-Hamiltonian, connected graph. For every pair of nonadjacent vertices u and v, 8(u) +8()2 k, for some k> O. Show that G contains a path of length k.
Long paths we show that for every n ≥ 3 if deg(v) ≥ n/2 for every v ∈ V then the graph contains a simple cycle (no vertex appears twice) that contains all vertices. Such a path is called an Hamiltonian path. From now on we assume that deg(v) ≥ n/2 for every v. 1. Show that the graph is connected (namely the distance between every two vertices is finite) 2. Consider the longest simple path x0, x1, . ....
2. Let G be an undirected graph. For every u,vE V(G), let dc(u,v) be the length of the shoertest path from u to v. The diameter of G is he maximum distance bet In other words: max (de(u, v) u,vEV(G) the running time of your algorithm 2. Let G be an undirected graph. For every u,vE V(G), let dc(u,v) be the length of the shoertest path from u to v. The diameter of G is he maximum distance bet In...
Let (u, v) be a minimum-weight edge in a connected graph G. Show that (u, v) belongs to some minimum spanning tree of G.
10) Shortest Paths (10 marks) Some pseudocode for the shortest path problem is given below. When DIJKSTRA (G, w,s) is called, G is a given graph, w contains the weights for edges in G, and s is a starting vertex DIJKSTRA (G, w, s) INITIALIZE-SINGLE-SOURCE(G, s) 1: RELAX (u, v, w) 1: if dlv] > dlu (u, v) then 2d[v] <- d[u] +w(u, v) 3 4: end if 4: while Q φ do 5: uExTRACT-MIN Q) for each vertex v...
graph G, let Bi(G) max{IS|: SC V(G) and Vu, v E S, d(u, v) 2 i}, 10. (7 points) Given a where d(u, v) is the length of a shortest path between u and v. (a) (0.5 point) What is B1(G)? (b) (1.5 points) Let Pn be the path with n vertices. What is B;(Pn)? (c) (2 points) Show that if G is an n-vertex 3-regular graph, then B2(G) < . Further- more, find a 3-regular graph H such that...
Reachability. You are given a connected undirected graph G = (V, E ) as an adjacency list. The graph G might not be connected. You want to fill-in a two-dimensional array R[,] so that R[u,v] is 1 if there is a path from vertex u to vertex v. If no such path exists, then R[u,v] is 0. From this two-dimensional array, you can determine whether vertex u is reachable from vertex v in O(1) time for any pair of vertices...
Let G be a DAG (a graph without loops) and u, v be two designated nodes (there are many other nodes in G). In particular, each node in G is labeled with a color and multiple nodes can share the same color. A good path is one where the number of green nodes is bigger than the number of yellow nodes. Describe an efficient algorithm to count the number of good paths from u to v.
Problem 8. (2+4+4 points each) A bipartite graph G = (V. E) is a graph whose vertices can be partitioned into two (disjoint) sets V1 and V2, such that every edge joins a vertex in V1 with a vertex in V2. This means no edges are within V1 or V2 (or symbolically: Vu, v E V1. {u, u} &E and Vu, v E V2.{u,v} &E). 8(a) Show that the complete graph K, is a bipartite graph. 8(b) Prove that no...
2. (a) Let G be a connected non-complete graph with order n 2 3 and diameter d. Prove that the connectivity K(G) of G satisfies d-1 (b) A connected graph is called unicyclic if it contains exactly one cycle. Prove that the edge-connectivity of any unicyclic graph is at most 2. 2. (a) Let G be a connected non-complete graph with order n 2 3 and diameter d. Prove that the connectivity K(G) of G satisfies d-1 (b) A connected...