Question

use fleurys algorithm to find an euler circuit.

complete the euler circuit

a. a, c, g, n, h, k, p, m, f, b, e, d, g, h, c, d, k, f, e, m, e, k, h, a
b. a, c, g, n, h, k, p, m, f, b, e, d, g, h, c, d, k, f, e, m, e, h, d, a
c. a, c, g, n, h, k, p, m, f, b, e, d, g, h, c, d, k, f, e, m, k, e, h, d, a
d. a, c, g, n, h, k, p, m, f, b, e, d, g, h, c, d, k, f, e, k, e, h, d, a

Explanation:

Step1: Recall Fleury's Algorithm

Fleury's Algorithm is used to find an Euler circuit in a graph. An Euler circuit is a closed walk that traverses every edge of the graph exactly once.

Step2: Analyze each option

• Option A:

Check the sequence of vertices. After \(M\), \(E\) is followed by \(K\). But in the graph structure (assuming standard graph - edge connections), after \(M\) (if we consider the path up to \(M\)), the next edge from \(E\) should be to \(M\) again (if following the circuit rules). So, this option is incorrect.

• Option B:

After \(M\), \(E\) is followed by \(H\). But if we consider the graph's edge - vertex connections (for an Euler circuit, we need to traverse all edges), the sequence \(M,F,B,E,D,G,H,C,D,K,F,E,M,E,H,D,A\) has a problem. When we get to \(E\) after \(M\), if we go to \(H\) from \(E\), we are not using all the edges incident to \(E\) in a proper circuit - forming way.

• Option C:

Check the vertex - edge sequence \(A,C,G,N,H,K,P,M,F,B,E,D,G,H,C,D,K,F,E,M,K,E,H,D,A\).

• Start at \(A\).
• Move \(A

ightarrow C\): uses an edge.

• \(C

ightarrow G\): uses an edge.

• \(G

ightarrow N\): uses an edge.

• \(N

ightarrow H\): uses an edge.

• \(H

ightarrow K\): uses an edge.

• \(K

ightarrow P\): uses an edge.

• \(P

ightarrow M\): uses an edge.

• \(M

ightarrow F\): uses an edge.

• \(F

ightarrow B\): uses an edge.

• \(B

ightarrow E\): uses an edge.

• \(E

ightarrow D\): uses an edge.

• \(D

ightarrow G\): uses an edge.

• \(G

ightarrow H\): uses an edge.

• \(H

ightarrow C\): uses an edge.

• \(C

ightarrow D\): uses an edge.

• \(D

ightarrow K\): uses an edge.

• \(K

ightarrow F\): uses an edge.

• \(F

ightarrow E\): uses an edge.

• \(E

ightarrow M\): uses an edge.

• \(M

ightarrow K\): uses an edge.

• \(K

ightarrow E\): uses an edge.

• \(E

ightarrow H\): uses an edge.

• \(H

ightarrow D\): uses an edge.

• \(D

ightarrow A\): uses an edge.

• Option D:

After \(F\), \(E\) is followed by \(K\) twice (\(E,K,E\)). But in an Euler circuit, each edge is traversed exactly once. So, this option is incorrect.

Answer:

C. A,C,G,N,H,K,P,M,F,B,E,D,G,H,C,D,K,F,E,M,K,E,H,D,A