Eulerian cycle is a cycle which starts and ends at the same graph vertex including all other vertices exactly once. In other words, it is a graph cycle which uses each graph edge exactly once Eulerian cycle in this graph is 1 -> 2 -> 4 -> 8 -> 5 -> 10 -> 12 -> 11 -> 9 -> 7 -> 6 -> 3 -> 1
A.) Prove that if some graph G is an Eulerian graph, the L(G) {the line graph of G} is also Eulerian. B.) Find a connected non-Eulerian graph for which the line graph is Eulerian.
Choose the true statement. If a graph G admits an Eulerian path, then G is connected. If a graph G admits an Eulerian path, then G admits a Hamiltonian path. If a graph G admits a Hamiltonian path, then G admits an Eulerian path. the four other possible answers are false If a graph G is connected, then G admits an Eulerian path.
Give a condition that is sufficient but not necessary for an undirected graph not to have an Eulerian Cycle. Justify your answer.
1. Give a condition that is necessary but not sufficient for an undirected graph to have an Eulerian Path. Justify your answer. 2. Give a condition that is sufficient but not necessary for an undirected graph not to have an Eulerian Cycle. Justify your answer.
A connected simple graph G has 16 vertices and 117 edges. Prove G is Hamiltonian and prove G is not Eulerian
Let G be a simple graph with at least four vertices. a) Give an example to show that G can contain a closed Eulerian trail, but not a Hamiltonian cycle. b) Give an example to show that G can contain a closed Hamiltonian cycle, but not a Eulerian trail.
Question 5: [10pt total] Let G be the following graph: True for False: Which of the following statements are true about G? 5)a) (1pt] G is a directed graph: 5)f) [1pt] G is bipartite: 5)b) [1pt] G is a weighted graph: 5)g) (1pt] G has a leaf vertex: ......... 5)c) [1pt] G is a multi-graph: 5)h) [1pt] G is planar: 5)d) [1pt] G is a loop graph: 5)i) [1pt] G is Eulerian: 5)) (1pt] G is a complete graph: 5)j)...
File Edit Format View Help Graphs and trees 4. [6 marks] Using the following graph representation (G(V,E,w)): v a,b,c,d,e,f E fa,b), (a,f),fa,d), (b,e), (b,d), (c,f),(c,d),(d,e),d,f)) W(a,b) 4,W(a,f) 9,W(a,d) 10 W(b,e) 12,W(b,d) 7,W(c,d) 3 a) Draw the graph including weights. b) Given the following algorithm for Inding a minimum spanning tree for a graph: Given a graph (G(V,E)) create a new graph (F) with nodes (V) and no edges Add all the edges (E) to a set S and order them...
consider the following problem: Given a Graph G = (V, E), does G have a cycle? Show that this problem is in NP.
5. (10 points) Solve TSP (Travelling Salesman Problem) for the following graph using 2-MST (Minimum Spanning Tree) algorithm. 18 12 15 15 13 10 15 Answer: a) the MST consists of edges its length is b) the Eulerian cycle is c) the Hamiltonian cycle is its length is