GATE CS 2014 Set 1 — Question 62
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Engineering Mathematics → Graph Theory (Math) → Graph Terminology (Degree, Paths, Cycles)
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Question
An ordered -tuple with is called graphic if there exists a simple undirected graph with vertices having degrees respectively. Which of the following 6-tuples is NOT graphic?
Correct answer
(C) (3, 3, 3, 1, 0, 0)
Solution
To determine if a sequence is graphic, we can use the Havel-Hakimi algorithm. The algorithm states that a sequence is graphic if and only if the sequence is graphic (after re-sorting if necessary).Let's test option (C) :
(A) : Sum is 6 (even). 3 disjoint edges. Graphic.
(B) : Sum is 12 (even). A cycle of 6 vertices (). Graphic.
(D) : Sum is 8 (even).
1.Remove the first element (). Subtract from the next elements.
Sequence becomes: .2.Sort the sequence: .
3.Remove the first element (). Subtract from the next elements.
Sequence becomes: .Since we have a negative degree (), the sequence is not graphic.Let's briefly check the others:(A) : Sum is 6 (even). 3 disjoint edges. Graphic.
(B) : Sum is 12 (even). A cycle of 6 vertices (). Graphic.
(D) : Sum is 8 (even).
- Remove 3: . Sort: .
- Remove 1: . Graphic.
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