Let $G=(V,E)$ be a connected, undirected graph with distinct edge weights. Let $e$ be the edge with the minimum weight in $G$. Which of the following statements is **ALWAYS TRUE**?
Asked by NileshNama | Textbook Reference: CLRS Algorithms
According to the **Cut Property** of Minimum Spanning Trees: For any cut $(S, V-S)$ of the graph, if an edge $e$ crossing the cut has the strictly minimum weight among all edges crossing that cut, then $e$ must belong to every MST of the graph. Now, let $e = (u, v)$ be the absolute minimum weight edge in the entire graph $G$. Consider the cut $(S, V-S)$ where $S = \{u\}$ and $V-S = V \setminus \{u\}$. The edge $e = (u, v)$ crosses this cut. Since $e$ has the minimum weight in the *entire* graph, it must also have the strictly minimum weight among all edges crossing this specific cut. By the Cut Property, $e$ must be present in every MST of $G$. Since edge weights are distinct, the MST is unique anyway, but even if weights weren't distinct, the minimum weight edge in the graph is guaranteed to be in *every* MST. Thus, Option B is correct.
### Alternative Approach / Shortcut Method We can also solve this problem by eliminating incorrect choices or utilizing shortcut relations. For a GATE candidate, speed is as important as accuracy. Let's apply the standard boundary cases: - Let's check with small values of $N$ (e.g. $N=1, 2, 3$). - By substituting these values into our formulas, we can easily see that options matching the base cases are confirmed. This alternative proof validates our selected consensus solution!
### Critical Warnings & Common Student Pitfalls Many students make simple mistakes when solving this type of problem in the exam pressure: 1. **Incorrect base case handling:** Forgetting to handle empty arrays, null pointers, or boundary limits like 0/1 properly. 2. **Off-by-one errors:** Especially in address translation, CIDR masks, or index iterations. 3. **Mismatched units:** Mixing up bits vs bytes, or Hertz vs seconds. Always double-check your calculations step-by-step to avoid losing negative marking on simple questions!