Let \( G \) be a simple, unweighted, and undirected graph. A subset of the vertices and edges of \( G \) are shown below.
It is given that \( a - b - c - d \) is a shortest path between \( a \) and \( d \); \( e - f - g - h \) is a shortest path between \( e \) and \( h \); \( a - f - c - h \) is a shortest path between \( a \) and \( h \). Which of the following is/are NOT the edges of \( G \)?
Let \( G \) be a simple, unweighted, and undirected graph. A subset of the vertices and edges of \( G \) are shown below.
It is given that \( a - b - c - d \) is a shortest path between \( a \) and \( d \); \( e - f - g - h \) is a shortest path between \( e \) and \( h \); \( a - f - c - h \) is a shortest path between \( a \) and \( h \). Which of the following is/are NOT the edges of \( G \)?
Show Hint
- \( (b,d) \)
- \( (b,g) \)
- \( (b,h) \)
- \( (e,g) \)
The Correct Option is A, C, D
Solution and Explanation
The shortest path from \( a \) to \( d \) is \( a - b - c - d \). This means there is no direct edge between \( b \) and \( d \).
The shortest path from \( e \) to \( h \) is \( e - f - g - h \). This implies that \( e \) and \( g \) are not directly connected.
The shortest path from \( a \) to \( h \) is \( a - f - c - h \), suggesting that \( b \) and \( h \) are not directly connected.
Top Questions on Graph Theory
- Consider the given graph and shortest paths: The shortest paths given are: \[ a - b - c - d, \quad e - f - g - h, \quad a - f - c - h \] Which of the following cannot be an edge in the original graph?
- The coefficient of \( x^{50} \) in \( (1 + x)^{101} (1 - x + x^2)^{100} \) is:
- VITEEE - 2024
- Mathematics
- Graph Theory
- The number of phone calls (in thousands) are made by a telephone company for five weeks as given below:Taking a period of moving averages as 3 weeks, the graph of moving averages can be depicted directly as :
Week \(1\) \(2\) \(3\) \(4\) \(5\) No. of Telephone calls \(110\) \(130\) \(93\) \(104\) \(211\) - CUET (UG) - 2023
- Mathematics
- Graph Theory
Questions Asked in GATE DA exam
Consider a directed graph \( G = (V,E) \), where \( V = \{0,1,2,\dots,100\} \) and
\[ E = \{(i,j) : 0 < j - i \leq 2, \text{ for all } i,j \in V \}. \] Suppose the adjacency list of each vertex is in decreasing order of vertex number, and depth-first search (DFS) is performed at vertex 0. The number of vertices that will be discovered after vertex 50 is:- GATE DA - 2025
- Data Structures
- Consider the following pseudocode.
Create empty stack S Set x = 0, flag = 0, sum = 0 Push x onto S while (S is not empty){ if (flag equals 0){ Set x = x + 1 Push x onto S } if (x equals 8): Set flag = 1 if (flag equals 1){ x = Pop(S) if (x is odd): Pop(S) Set sum = sum + x } } Output sumThe value of \( sum \) output by a program executing the above pseudocode is:- GATE DA - 2025
- Data Structures
- Consider the following Python code snippet.
def f(a, b): if (a == 0): return b if (a % 2 == 1): return 2 * f((a - 1) / 2, b) return b + f(a - 1, b) print(f(15, 10))The value printed by the code snippet is 160 (Answer in integer).- GATE DA - 2025
- Recursion
Consider the following tables, Loan and Borrower, of a bank.
Query: \[ \pi_{\text{branch\_name}, \text{customer\_name}} (\text{Loan} \bowtie \text{Borrower}) \div \pi_{\text{branch\_name}}(\text{Loan}) \] where \( \bowtie \) denotes natural join. The number of tuples returned by the above relational algebra query is 1 (Answer in integer).- GATE DA - 2025
- Relational Algebra
- Let \( D = \{ x^{(1)}, x^{(2)}, \dots, x^{(n)} \} \) be a dataset of \( n \) observations where each \( x^{(i)} \in \mathbb{R}^{100} \). It is given that
\[ \sum_{i=1}^{n} x^{(i)} = 0. \] The covariance matrix computed from \( D \) has eigenvalues \( \lambda_i = 100^2 - i \), for \( 1 \leq i \leq 100 \). Let \( u \in \mathbb{R}^{100} \) be the direction of maximum variance with \( u^T u = 1 \). The value of
\[ \frac{1}{n} \sum_{i=1}^{n} \left( u^T x^{(i)} \right)^2 = \underline{\hspace{2cm}} \quad \text{(Answer in integer)} \]- GATE DA - 2025
- Matrix algebra