Question

modify the graph by removing the least number of edges so that the resulting graph has an euler circuit. then find an euler circuit for the modified graph.
which edge(s) should be removed so that the resulting graph has an euler circuit?
(use a comma to separate answers as needed.)

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

• Degree of \(A\): \(2\) (even)
• Degree of \(B\): \(4\) (even)
• Degree of \(C\): \(3\) (odd)
• Degree of \(D\): \(3\) (odd)
• Degree of \(E\): \(1\) (odd)
• Degree of \(F\): \(2\) (even)
• Degree of \(G\): \(3\) (odd)
• Degree of \(H\): \(3\) (odd)

Step3: Determine edges to remove

We need to make all vertex degrees even. Removing edges \(CE\) and \(DG\) (or other equivalent pairs that adjust the odd - degree vertices to even) will work. For example, if we remove \(CE\) and \(DG\):

• Degree of \(C\) becomes \(2\) (was \(3\)), degree of \(E\) becomes \(0\) (but graph is still connected as there are other paths), degree of \(D\) becomes \(2\) (was \(3\)), degree of \(G\) becomes \(2\) (was \(3\)).

Answer:

\(CE,DG\)