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,k,e,h,d,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,e,k,h,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: Understand 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 a graph exactly once. The algorithm states that we should start at a vertex (in a connected graph where all vertices have even degrees, which is the case for a graph with an Euler circuit). Then, at each step, we choose an edge to traverse that is not a bridge (an edge whose removal increases the number of connected components of the graph) unless there is no other choice.

Step2: Analyze each option

• Option A: Check the sequence. An Euler circuit must start and end at the same vertex. Also, each edge should be traversed exactly once. In this option, the sequence \(A,C,G,N,H,K,P,M,F,B,E,D,G,H,C,D,K,F,E,M,K,E,H,D,A\) has a problem. The sub - sequence \(M,K,E,H,D,A\) after \(E,M,K,E,H,D,A\) has an incorrect edge traversal. The edge \(KE\) is traversed more than once in an incorrect order.
• Option B: Consider the sequence \(A,C,G,N,H,K,P,M,F,B,E,D,G,H,C,D,K,F,E,M,E,H,D,A\). Start at \(A\). Traverse edges as per the sequence. Each edge is traversed exactly once, and we return to \(A\) at the end.
• Option C: Look at the sequence \(A,C,G,N,H,K,P,M,F,B,E,D,G,H,C,D,K,F,E,M,E,K,H,A\). The ending \(E,K,H,A\) has an incorrect edge traversal. The vertex \(K\) is not connected back to \(A\) in a valid non - bridge - violating way after the previous traversals.
• Option D: Examine the sequence \(A,C,G,N,H,K,P,M,F,B,E,D,G,H,C,D,K,F,E,K,E,H,D,A\). The sub - sequence \(E,K,E\) has an edge \(KE\) traversal that is incorrect (edges should be traversed exactly once in an Euler circuit).

Answer:

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