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Let D be a simple graph on 10 vertices such that there is a vertex of degree 1, a vertex of degree 2, a vertex of degree 3, a vertex of degree 4, a vertex of degree 5, a vertex of degree 6, a vertex of degree 7, a vertex of degree 8 and a vertex of degree 9. What can be the degree of the last vertex?
Let G be a directed graph whose vertex set is the set of numbers from 1 to 100. There is an edge from a vertex i to a vertex j if and only if either j = i + 1 or j = 3i. The minimum number of edges in a path in G from vertex 1 to vertex 100 is 
Let G be a directed graph whose vertex set is the set of numbers from 1 to 50. There is an edge from a vertex i to a vertex j if and only if either j = i + 1 or j = 3i. Calculate the minimum number of edges in a path in G from vertex 1 to vertex 50.
Consider the directed graph shown in the figure below. There are multiple shortest paths between vertices S and T. Which one will be reported by Dijkstra’s shortest path algorithm? Assume that, in any iteration, the shortest path to a vertex v is updated only when a strictly shorter path to v is discovered.
The line graph L(G) of a simple graph G is defined as follows: There is exactly one vertex v(e) in L(G) for each edge e in G. For any two edges e and e' in G, L(G) has an edge between v(e) and v(e'), if and only if e and e' are incident with the same vertex in G. Which of the following statements is/are TRUE? (P) The line graph of a cycle is a cycle. (Q) The line graph of a clique is a clique. (R) The line graph of a planar graph is planar. (S) The line graph of a tree is a tree.
Garima starts walking towards the south from point R. He walks for 5m to reach Point S then turns right and walks for 2m to reach point T from there he turns left and walks for 3m to reach Point Z. He then turns right from point Z and walks for 6m to reach point Y and then turns right and walks for 3m to reach point X. He then turns right from point X and walks for 4m to reach point W and then turns left and walks for 5m to reach point V and then turns left and walks for 4m till point U and stops. What is the shortest distance between point V and point R?
A man starts from point ‘O’, travels 20 km towards east to reach point ‘A’, turns right and travels 10 km to reach point ‘B', turns right and travels 9 km to reach point ‘C’, turns right and travels 5 km to reach point ‘D’, turns left and travels 12 km to reach point ‘E’ and then turns right and travels 6 km to reach point ‘F. What is the shortest distance between point ‘E’ and point ‘C’?
A man starts from point 'O', travels 20 km towards east to reach point 'A', turns right and travels 10 km to reach point 'B', turns right and travels 9 km to reach point 'C', turns right and travels 5 km to reach point 'D', turns left and travels 12 km to reach point 'E' and then turns right and travels 6 km to reach point 'F'. What is the shortest distance between point 'E' and point 'C'?
Consider a network with nodes 1, 2, 3, 4, 5 and 6. The nodes are connected with directed arcs as shown in the table below. The respective costs (in INR) incurred while traversing the directed arcs are also mentioned. Directed arcs  1 → 2 1 → 3 2 → 4 2 → 5 3 → 2 3 → 4 3 → 5 4 → 5 4 → 6 5 → 6 Cost (in INR) 3 9 3 2 2 4 8 7 2 2 The second shortest path from node 1 to node 6 (i.e. the path that has the second least total cost and does not use any part of the shortest path) has a total cost (in INR) of
Which of the following statements are comes for Network Critical Path? 1. The path of critical activities, which links the start and end events is critical path 2. It is the path of activities having zero float 3. It is the path of events having zero slack 4. The sum of the duration of the critical activities along a critical path gives the duration of the Project