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Consider the following statement about the planarity of the graph.
(1) The number of regions (r) in a graph G (V, E) can be given by Euler formula r = E + V – 2
(2) If G is a connected planar simple graph with e edges and v vertices, where v >= 3 then
(e <= 3v="" –="">
(3) If in a planar graph G (v, e) there is a cycle with 4 or more vertices then (e <= 2v="" –="">
(4) Bipartite can’t have odd length cycles.
The number of correct statements is ________________?
Correct answer is '3'. Can you explain this answer?
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Verified Answer
Consider the following statement about the planarity of the graph.(1)...
Statement 1 is false, because the Euler formula is r = E – V + 2.
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Statement 2 and statement 3 are true, they are the corollary of the Euler formula. To get these, for statement 2, put (2e/3) >= r in the Euler formula.
To get statement 3, put (2e / 4) >= r, as the degree of every region will be greater than 4.
Statement 4 is also true. Just try to draw a bipartite graph that is having odd length cycle. And you will understand it.
Most Upvoted Answer
Consider the following statement about the planarity of the graph.(1)...
Understanding the Statements
To evaluate the correctness of the statements about graph planarity, let's break down each one:
Statement 1: Euler's Formula
- The number of regions (r) in a planar graph is given by Euler's formula, which states: r = E + V - F, where F is the number of faces.
- This statement is incorrect as it misrepresents the formula by substituting F with 2.
Statement 2: Edge Condition for Planar Graphs
- For a connected planar simple graph with v vertices (v ≥ 3), the formula e ≤ 3v - 6 holds for the maximum number of edges (e).
- This statement is correct, as it adheres to the established bounds on edges in planar graphs.
Statement 3: Cycle Condition
- If a planar graph G has a cycle with 4 or more vertices, it implies that there can be multiple edges, but it does not directly affect the edge count condition.
- This statement is also correct, as it indirectly supports the maximum edge condition stated in Statement 2.
Statement 4: Bipartite Graphs
- It is true that bipartite graphs cannot have odd-length cycles.
- This statement is correct and aligns with the definition of bipartite graphs.
Summary of Correctness
- Statement 1: Incorrect
- Statement 2: Correct
- Statement 3: Correct
- Statement 4: Correct
Thus, the total number of correct statements is 3.
To evaluate the correctness of the statements about graph planarity, let's break down each one:
Statement 1: Euler's Formula
- The number of regions (r) in a planar graph is given by Euler's formula, which states: r = E + V - F, where F is the number of faces.
- This statement is incorrect as it misrepresents the formula by substituting F with 2.
Statement 2: Edge Condition for Planar Graphs
- For a connected planar simple graph with v vertices (v ≥ 3), the formula e ≤ 3v - 6 holds for the maximum number of edges (e).
- This statement is correct, as it adheres to the established bounds on edges in planar graphs.
Statement 3: Cycle Condition
- If a planar graph G has a cycle with 4 or more vertices, it implies that there can be multiple edges, but it does not directly affect the edge count condition.
- This statement is also correct, as it indirectly supports the maximum edge condition stated in Statement 2.
Statement 4: Bipartite Graphs
- It is true that bipartite graphs cannot have odd-length cycles.
- This statement is correct and aligns with the definition of bipartite graphs.
Summary of Correctness
- Statement 1: Incorrect
- Statement 2: Correct
- Statement 3: Correct
- Statement 4: Correct
Thus, the total number of correct statements is 3.
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Consider the following statement about the planarity of the graph.(1) The number of regions (r) in a graph G (V, E) can be given by Euler formula r = E + V – 2(2) If G is a connected planar simple graph with e edges and v vertices, where v >= 3 then(e (3) If in a planar graph G (v, e) there is a cycle with 4 or more vertices then (e (4) Bipartite can’t have odd length cycles.The number of correct statements is ________________?Correct answer is '3'. Can you explain this answer?
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Consider the following statement about the planarity of the graph.(1) The number of regions (r) in a graph G (V, E) can be given by Euler formula r = E + V – 2(2) If G is a connected planar simple graph with e edges and v vertices, where v >= 3 then(e (3) If in a planar graph G (v, e) there is a cycle with 4 or more vertices then (e (4) Bipartite can’t have odd length cycles.The number of correct statements is ________________?Correct answer is '3'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider the following statement about the planarity of the graph.(1) The number of regions (r) in a graph G (V, E) can be given by Euler formula r = E + V – 2(2) If G is a connected planar simple graph with e edges and v vertices, where v >= 3 then(e (3) If in a planar graph G (v, e) there is a cycle with 4 or more vertices then (e (4) Bipartite can’t have odd length cycles.The number of correct statements is ________________?Correct answer is '3'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statement about the planarity of the graph.(1) The number of regions (r) in a graph G (V, E) can be given by Euler formula r = E + V – 2(2) If G is a connected planar simple graph with e edges and v vertices, where v >= 3 then(e (3) If in a planar graph G (v, e) there is a cycle with 4 or more vertices then (e (4) Bipartite can’t have odd length cycles.The number of correct statements is ________________?Correct answer is '3'. Can you explain this answer?.
Consider the following statement about the planarity of the graph.(1) The number of regions (r) in a graph G (V, E) can be given by Euler formula r = E + V – 2(2) If G is a connected planar simple graph with e edges and v vertices, where v >= 3 then(e (3) If in a planar graph G (v, e) there is a cycle with 4 or more vertices then (e (4) Bipartite can’t have odd length cycles.The number of correct statements is ________________?Correct answer is '3'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider the following statement about the planarity of the graph.(1) The number of regions (r) in a graph G (V, E) can be given by Euler formula r = E + V – 2(2) If G is a connected planar simple graph with e edges and v vertices, where v >= 3 then(e (3) If in a planar graph G (v, e) there is a cycle with 4 or more vertices then (e (4) Bipartite can’t have odd length cycles.The number of correct statements is ________________?Correct answer is '3'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statement about the planarity of the graph.(1) The number of regions (r) in a graph G (V, E) can be given by Euler formula r = E + V – 2(2) If G is a connected planar simple graph with e edges and v vertices, where v >= 3 then(e (3) If in a planar graph G (v, e) there is a cycle with 4 or more vertices then (e (4) Bipartite can’t have odd length cycles.The number of correct statements is ________________?Correct answer is '3'. Can you explain this answer?.
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