GATE CS 2014 Set 3 Q.61 (CS Q.51): Graph Theory

If $G$ is a forest with $n$ vertices and $k$ connected components, how many edges does $G$ have?

$\lfloor n/k \rfloor$

$\lceil n/k \rceil$

$n - k$

$n - k + 1$

Official answer and solution

Solution

Sum the edge counts across all connected components of an acyclic graph.

A forest is defined as an undirected acyclic graph.
Each connected component of a forest is a tree.
Let $k$ be the number of connected components, and let $n_i$ denote the number of vertices in component $i$ for $1 \le i \le k$ .
- The sum of vertices over all components is: $\sum_{i=1}^k n_i = n.$
For any tree with $n_i$ vertices, the number of edges is exactly $n_i - 1$ .
Summing edges across all $k$ trees:
$\text{Total Edges} = \sum_{i=1}^k (n_i - 1) = \sum_{i=1}^k n_i - \sum_{i=1}^k 1 = n - k.$

This matches option C ( $n - k$ ).

Final answer

Common trap

Using the tree formula n - 1 instead of subtracting k for a forest with k components.

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