GATE CS Minimum Spanning Trees questions

Let $G(V, E)$ be a simple, undirected, edge-weighted graph with unique edge weights. Which of the following statements about the minimum spanning trees (MST) of $G$ is/are true?

Let $G$ be any undirected graph with positive edge weights, and $T$ be a minimum spanning tree of $G$. For any two vertices, $u$ and $v$, let $d_{1}(u, v)$ and $d_{2}(u, v)$ be the shortest distances between $u$ and $v$ in $G$ and $T$, respectively. Which ONE of the options is CORRECT for all possible $G, T, u$ and $v$ ?

The maximum value of $\text{x}$ such that the edge between the nodes $\text{B}$ and $\text{C}$ is included in every minimum spanning tree of the given graph is _________. (answer in integer)

​​​​Let $\text{G}$ be an edge-weighted undirected graph with positive edge weights. Suppose a positive constant $\alpha$ is added to the weight of every edge. Which ONE of the following statements is TRUE about the minimum spanning trees (MSTs) and shortest paths (SPs) in $\text{G}$ before and after the edge weight update?

The number of distinct minimum-weight spanning trees of the following graph is

Consider a simple undirected weighted graph $\textit{G},$ all of whose edge weights are distinct. Which of the following statements about the minimum spanning trees of $\textit{G}$ is/are $\text{TRUE}?$

Let $\textit{G(V,E)}$ be a directed graph, where $\textit{V} = \{ 1, 2, 3, 4, 5 \}$ is the set of vertices and $\textit{E}$ is the set of directed edges, as defined by the following adjacency matrix $\textit{A}.$ $$A [i][j] = \left\{\begin{matrix} 1, & 1 \leq j \leq i \leq 5 \\ 0, & \text{otherwise}\end{matrix}\right.$$ $A[i][j] = 1$ …

Consider the following undirected graph with edge weights as shown: The number of minimum-weight spanning trees of the graph is ___________.

Let $G$ be a connected undirected weighted graph. Consider the following two statements. $S_1$: There exists a minimum weight edge in $G$ which is present in every minimum spanning tree of $G$. $S_2$: If every edge in $G$ has distinct weight, then $G$ has a unique minimum spanning tree. Which one of the following options is correct?

Consider the following directed graph: Which of the following is/are correct about the graph?

Let $G = (V, E)$ 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 $(u, v) \in V \times V$ is added to $G$. The worst case time complexity of determining if $T$ is still an MST of the resultant graph is

Consider a graph $G = (V,E)$, where $V = \{v_1,v_2, \dots ,v_{100}\}$, $E = \{(v_i,v_j) \mid 1\leq i < j \leq 100\}$, and weight of the edge $(v_i,v_j)$ is $\mid i – j \mid$. The weight of minimum spanning tree of $G$ is _________

Consider the following undirected graph $G$: Choose a value for $x$ that will maximize the number of minimum weight spanning trees (MWSTs) of $G$. The number of MWSTs of $G$ for this value of $x$ is ____.

$G=(V, E)$ is an undirected simple graph in which each edge has a distinct weight, and $e$ is a particular edge of $G$. Which of the following statements about the minimum spanning trees $(MSTs)$ of $G$ is/are TRUE? If $e$ is the lightest edge of some cycle in $G$, then every MST of $G$ includes $e$. If $e$ is the heaviest edge of some …

Let $G$ be a complete undirected graph on $4$ vertices, having $6$ edges with weights being $1, 2, 3, 4, 5,$ and $6$. The maximum possible weight that a minimum weight spanning tree of $G$ can have is __________

Let $G$ be a weighted connected undirected graph with distinct positive edge weights. If every edge weight is increased by the same value, then which of the following statements is/are TRUE? $P$: Minimum spanning tree of $G$ does not change. $Q$: Shortest path between any pair of vertices does not change.

The graph shown below has $8$ edges with distinct integer edge weights. The minimum spanning tree (MST) is of weight $36$ and contains the edges: $\{(A, C), (B, C), (B, E), (E, F), (D, F)\}$. The edge weights of only those edges which are in the MST are given in the figure shown below. The minimum possible sum of weights of all $8$ edges of this …

Let $G$ be a connected undirected graph of $100$ vertices and $300$ edges. The weight of a minimum spanning tree of $G$ is $500$. When the weight of each edge of $G$ is increased by five, the weight of a minimum spanning tree becomes ______.

The number of distinct minimum spanning trees for the weighted graph below is _____

Let $G$ be a weighted graph with edge weights greater than one and $G'$ be the graph constructed by squaring the weights of edges in $G$. Let $T$ and $T'$ be the minimum spanning trees of $G$ and $G'$, respectively, with total weights $t$ and $t'$. Which of the following statements is TRUE?