Question

draw a graph with the given characteristics. the graph has four even vertices. choose the correct answer below. \\(\bigcirc\\) a. graph with vertices a, b, c, d \\(\bigcirc\\) b. graph with vertices a, b, c, d \\(\bigcirc\\) c. graph with vertices a, b, c, d \\(\bigcirc\\) d. graph with vertices a, b, c, d \\(\bigcirc\\) e. such a graph is not possible

Explanation:

Step1: Recall the definition of an even vertex

A vertex is even if the number of edges incident to it is even.

Step2: Analyze each option

• Option A:
• Vertex \(B\) has degree \(1\) (odd). So this option is incorrect.
• Option B:
• Calculate the degree of each vertex:
• \(\text{deg}(A) = 2\), \(\text{deg}(B)=2\), \(\text{deg}(C)=2\), \(\text{deg}(D)=2\). All vertices have even degrees.
• Option C:
• Vertex \(D\) has degree \(3\) (odd). So this option is incorrect.
• Option D:
• Vertex \(B\) has degree \(3\) (odd). So this option is incorrect.
• Option E:
• Since option \(B\) is valid, this option is incorrect.

Answer:

B.