Question

a graph is given to the right
a. modify the graph by removing the least number of edges so that the resulting graph has an euler circuit
b. find an euler circuit for the modified graph

a. which edge(s) should be removed so that the resulting graph has an euler circuit?
bf,ei (use a comma to separate answers as needed.)
b. there are several euler circuits for this graph that begin at vertex a. choose the answer below that describes one such euler circuit.
○ a. a,d,i,h,i,d,c,b,g,j,h,g,f,c,a
○ b. a,c,f,g,b,c,d,e,h,j,g,b,c,a
○ c. a,d,e,h,i,d,c,b,g,f,c,a
○ d. a,c,b,g,f,c,d,e,h,j,g,h,i,d,a

Explanation:

Step1: Recall Euler circuit condition

An Euler circuit exists in a connected graph if and only if every vertex has an even degree.

Step2: Analyze vertex degrees (original graph)

Count degrees of vertices. Removing edges \(BF\) and \(EI\) makes all vertices have even degrees.

Step3: Check Euler circuit path

For option D:

• Start at \(A\).
• \(A - C - B - G - F - C - D - E - H - J - G - H - I - D - A\) traverses all edges exactly once and returns to \(A\).

Answer:

a. \(BF, EI\)
b. D. \(ACB,GFC,DEH,JGH,IDA\)