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The time complexity of Prim's algorithm depends on the data structures used. Using an adjacency matrix with searching gives $O(V^2)$, while using a binary heap with an adjacency list gives $O(V \log V + E \log V)$, where $V$ is the number of vertices and $E$ is the number of edges.

Conditions: The graph is represented using specific data structures.; $V$ denotes the number of vertices.; $E$ denotes the number of edges.

The time complexity of Prim's algorithm depends on the data structures used. Using an adjacency matrix with searching gives $O(V^2)$, while using a binary heap with an adjacency list gives $O(V \log V + E \log V)$, where $V$ is the number of vertices and $E$ is the number of edges.

Conditions: The graph is represented using specific data structures.; $V$ denotes the number of vertices.; $E$ denotes the number of edges.

A minimum spanning tree (MST) is a connected, acyclic subgraph that spans all vertices of a weighted graph and has the minimum possible total edge weight.

Conditions: The underlying graph must be weighted.; The graph must be connected for a spanning tree to exist.; The MST minimizes the sum of selected edge weights.; The MST contains no cycles.

A minimum spanning tree (MST) is a connected, acyclic subgraph that spans all vertices of a weighted graph and has the minimum possible total edge weight.

Conditions: The underlying graph must be weighted.; The graph must be connected for a spanning tree to exist.; The MST minimizes the sum of selected edge weights.; The MST contains no cycles.