Network in Graph Theory · Form 4
Network in Graph Theory: Common Mistakes
The mistakes that quietly cost marks in Network in Graph Theory, and how to avoid each one in the SPM exam.
In our experience teaching Network in Graph Theory, most lost marks come from a small set of repeated slips, not from a lack of understanding. Here they are, with the fix for each.
Mistakes to avoid
- Miscounting degree when edges are drawn too close together
- Ignoring the direction of a directed edge
- Leaving a network diagram messy so a marker cannot follow it
Six more slips that quietly cost marks
- What students write: a loop at vertex P adds 1 to its degree. → Why it loses marks: a loop touches the vertex at both ends, so it counts twice. → Correct working: count the loop as 2, so if P has 3 other edges its degree is 3 + 2 = 5.
- What students write: two separate roads joining A and B are one edge. → Why it loses marks: each line drawn is its own edge, so both count. → Correct working: count 2 edges between A and B; each adds 1 to the degree of A and of B.
- What students write: guess the last vertex's degree by eye. → Why it loses marks: a guess is not a method and is often wrong. → Correct working: use total degrees = 2 × edges. With edges = 8 the total is 16; subtract the known degrees to get the missing one.
- What students write: a network with 6 points has 6 edges. → Why it loses marks: vertices and edges are counted separately and rarely match. → Correct working: count the points for vertices and the lines for edges; 6 vertices may join with any number of edges, so count the lines.
- What students write: any connected network is a tree. → Why it loses marks: a tree must have no cycle and exactly (vertices − 1) edges. → Correct working: for 5 vertices a tree has 4 edges; if a connected graph has 5 edges it contains a cycle and is not a tree.
- What students write: total a route by adding the vertex labels. → Why it loses marks: a route's total is the sum of the edge weights along it, not the vertices. → Correct working: for P → Q → R with edges 6 and 4, the route weight is 6 + 4 = 10, whatever the vertices are called.
The costliest slip to avoid
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
I got an odd total when I added all the degrees, is that possible?
No. The total of every vertex's degree equals twice the number of edges, and any number doubled is even.
An odd total means you miscounted somewhere, most often a loop counted as 1 instead of 2, or an edge you missed. Recount before you trust any later answer.
I counted a loop as one, how much does that cost?
More than one mark, usually. The wrong degree flows into the degree total, any missing-degree calculation, and any check you do, so a single loop error can wrong three linked answers.
Fix the habit now: a loop always adds 2, because it meets its vertex at both ends.
Two towns are joined by two separate roads, is that one edge or two?
Two. Every line drawn between the towns is a separate edge, called parallel edges, and each one raises the degree of both towns by one.
Count the lines you see, not the pairs of towns. The same holds when a table lists the same link twice for a genuine reason.