Top MCQs on Graph Theory in Mathematics

Question 81

A certain tree has two vertices of degree 4, one vertex of degree 3 and one vertex of degree 2. If the other vertices have degree 1, how many vertices are there in the graph?

• 5
• n - 3
• 20
• 11

Question 82

The number of edges in a regular graph of degree d and n vertices is

• maximum of n and d
• n + d
• nd
• nd / 2

Question 83

E is the number of edges in the graph and f is maximum flow in the graph. When the capacities are integers, the runtime of Ford-Fulberson algorithm is bounded by :

• O (E∗f)
• O (E² ∗f)
• O (E∗f² )
• O (E² ∗f² )

Question 84

Let G=(V,E) be a directed, weighted graph with weight function w:E→R. For some function f:V→R, for each edge (u,v)∈E, define w′(u,v) as w(u,v)+f(u)−f(v). Which one of the options completes the following sentence so that it is TRUE ? “The shortest paths in G under w are shortest paths under w′ too, _________”.

• for every f:V→R
• if and only if ∀u∈V, f(u) is positive
• if and only if ∀u∈V, f(u) is negative
• if and only if f(u) is the distance from s to u in the graph obtained by adding a new vertex s to G and edges of zero weight from s to every vertex of G

Question 85

Graph G is obtained by adding vertex s to K_3,4 and making s adjacent to every vertex of K_3,4. The minimum number of colours required to edge-colour G is _________ . Note - This question was Numerical Type.

• 2

• 3

• 5

• 7

Question 86

Let G=(V,E) be an undirected unweighted connected graph. The diameter of G is defined as:

Let M be the adjacency matrix of G. 
Define graph G2 on the same set of vertices with adjacency matrix N, where

Which one of the following statements is true?

• diam(G₂)≤⌈ diam(G)/2⌉

• ⌈ diam(G)/2⌉2)< diam(G)
  

• diam(G₂) = diam(G)

• diam(G)< diam(G₂)≤2 diam(G)

Question 87

Consider a weighted undirected graph with positive edge weights and let uv be an edge in the graph. It is known that the shortest path from the source vertex s to u has weight 53 and the shortest path from s to v has weight 65. Which one of the following statements is always true?

• weight (u, v) < 12
• weight (u, v) ≤ 12
• weight (u, v) > 12
• weight (u, v) ≥ 12

Question 88

Consider a full binary tree with n internal nodes, internal path length i, and external path length e. The internal path length of a full binary tree is the sum, taken over all nodes of the tree, of the depth of each node. Similarly, the external path length is the sum, taken over all leaves of the tree, of the depth of each leaf. Which of the following is correct for the full binary tree?

• e = i+n
• e = i+2n
• e = 2i+n
• e = 2ⁿ+i

Question 89

Let G = (V, G) be a weighted undirected graph and let T be a Minimum Spanning Tree (MST) of G maintained using adjacency lists. Suppose a new weighed edge (u, v) ∈ V×V is added to G. The worst case time complexity of determining if T is still an MST of the resultant graph is

• Θ(∣E∣ + ∣V∣)
• Θ(∣E∣.∣V∣)
• Θ(E∣ log ∣V∣)
• Θ(∣V∣)

Question 90

Suppose that a connected planar graph has six vertices, each of degrees four. Into how many regions is the plane divided by a planar representation of this graph?

• 6
• 8
• 12
• 10