Quick questions on Minimum spanning trees and Prim's algorithm for HSC Maths Standard 2

A spanning tree of a connected graph is a subgraph that:

A minimum spanning tree is a spanning tree with the smallest total edge weight among all possible spanning trees of the graph.

Kruskal's algorithm sorts all edges and adds them in order, skipping any that create a cycle:

The HSC accepts either; show your working clearly so the marker can follow which algorithm you used.

An edge creates a cycle if both endpoints are already connected (directly or via existing tree edges). For small graphs you can spot this visually; for larger graphs, keep track of which vertices are in which component as you add edges.

Sort all edges by weight: $AC(2), CD(3), AB(4), BC(5), DE(6), CE(7), BD(10)$.

A regional NSW council wants to lay fibre optic cable connecting $6$ small towns. The possible routes follow existing roads with known lengths. Total available routes: $9$ edges between the $6$ town vertices. The MST gives the minimum total cable needed: $6 - 1 = 5$ edges, total length determined by the specific weights.

The cheapest unused edge sometimes creates a cycle. Skip those.

Markers want to see each step: which edge added, which considered and skipped. A bare final tree without working loses marks.

Sum all the chosen edge weights at the end.

A spanning tree must connect every vertex. If your $n - 1$ edges form a forest (disconnected components), you have made a mistake. :::