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Let d denote the minimum degree of a vertex in a graph. For all planar graphs on n vertices with d ≥ 3, which one of the following is TRUE?
  • a)
    In any planar embedding, the number of faces is at least n/2 + 2
  • b)
    In any planar embedding, the number of faces is less than n/2 + 2
  • c)
    There is a planar embedding in which the number of faces is less than n/2 + 2
  • d)
    There is a planar embedding in which the number of faces is at most n/(d+1)
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Let d denote the minimum degree of a vertex in a graph. For all planar...
Euler's formula for planar graphs:
v − e + f = 2.
v => Number of vertices
e => Number of edges
f => Number of faces
Since degree of every vertex is at least 3, below is true from handshaking lemma (Sum of degrees is twice the number of edges)
3v >= 2e
3v/2 >= e
Putting these values in Euler's formula.
v - 3v/2 + f >= 2
f >= v/2 + 2
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Let d denote the minimum degree of a vertex in a graph. For all planar graphs on n vertices with d ≥ 3, which one of the following is TRUE?a)In any planar embedding, the number of faces is at least n/2 + 2b)In any planar embedding, the number of faces is less than n/2 + 2c)There is a planar embedding in which the number of faces is less than n/2 + 2d)There is a planar embedding in which the number of faces is at most n/(d+1)Correct answer is option 'A'. Can you explain this answer?
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Let d denote the minimum degree of a vertex in a graph. For all planar graphs on n vertices with d ≥ 3, which one of the following is TRUE?a)In any planar embedding, the number of faces is at least n/2 + 2b)In any planar embedding, the number of faces is less than n/2 + 2c)There is a planar embedding in which the number of faces is less than n/2 + 2d)There is a planar embedding in which the number of faces is at most n/(d+1)Correct answer is option 'A'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Let d denote the minimum degree of a vertex in a graph. For all planar graphs on n vertices with d ≥ 3, which one of the following is TRUE?a)In any planar embedding, the number of faces is at least n/2 + 2b)In any planar embedding, the number of faces is less than n/2 + 2c)There is a planar embedding in which the number of faces is less than n/2 + 2d)There is a planar embedding in which the number of faces is at most n/(d+1)Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let d denote the minimum degree of a vertex in a graph. For all planar graphs on n vertices with d ≥ 3, which one of the following is TRUE?a)In any planar embedding, the number of faces is at least n/2 + 2b)In any planar embedding, the number of faces is less than n/2 + 2c)There is a planar embedding in which the number of faces is less than n/2 + 2d)There is a planar embedding in which the number of faces is at most n/(d+1)Correct answer is option 'A'. Can you explain this answer?.
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