Exercise 1: Kruskal's Algorithm on P-Q-R-S — Possible Solution
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GIVEN
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Edges: P-Q(3), P-R(1), Q-R(4), Q-S(2), R-S(5)

STEP 1: SORT EDGES BY WEIGHT, ASCENDING
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P-R(1), Q-S(2), P-Q(3), Q-R(4), R-S(5)

STEP 2: PROCESS EACH EDGE IN ORDER
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P-R(1): P and R are in different components (each still its own
component) -> ADD. Components now: {P,R}, {Q}, {S}

Q-S(2): Q and S are in different components -> ADD.
Components now: {P,R}, {Q,S}

P-Q(3): P is in {P,R}, Q is in {Q,S} - different components ->
ADD. Components now: {P,R,Q,S} (all one component)

Q-R(4): Q and R are now BOTH in the same merged component ->
SKIP (would create a cycle)

R-S(5): R and S are now BOTH in the same merged component ->
SKIP (would create a cycle)

RESULT
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MST edges: P-R(1), Q-S(2), P-Q(3)
Total weight: 1 + 2 + 3 = 6

This uses exactly 3 edges for 4 vertices (V-1 = 3), matching this
chapter's own spanning-tree definition, and connects every vertex
- P and R directly, Q and S directly, and the two pairs joined
together through P-Q.

WHY THIS WORKS AS AN ANSWER
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Every edge is checked against the Union-Find components exactly
as this chapter's own method specifies - the two skipped edges are
correctly identified as connecting vertices already joined through
earlier additions, not merely because they were expensive.