To solve the problem of finding the Euler circuit of the graph using Fleury's algorithm, here is the step-by-step explanation:

Fleury's Algorithm Steps:

1. Check if the graph has an Euler circuit:

   • A graph has an Euler circuit if and only if it is connected and all vertices have even degrees.
   • From the graph, ensure all vertices meet this requirement.

2. Start at any vertex:

   • Since the graph is connected and all vertices have even degrees, we can start at any vertex.

3. Follow edges one at a time:

   • Choose an edge that does not leave the graph disconnected unless it is the only option. In Fleury's algorithm, avoid bridges unless they are necessary.

4. Remove edges as you traverse them:

   • Each edge used is "removed" from the graph, and you continue until all edges are traversed.

5. Ensure all edges are used once:

   • The traversal stops when you return to the starting vertex and all edges have been used exactly once.

Solution:

1. Check degrees of the vertices:

   • Degree of $a$: 2
   • Degree of $b$: 4
   • Degree of $c$: 4
   • Degree of $d$: 4
   • Degree of $e$: 2
   • Degree of $f$: 2
   • Degree of $g$: 2
   • Degree of $h$: 2
   • All vertices have even degrees, so the graph has an Euler circuit.

2. Start the Euler circuit:

   • Start at any vertex (e.g., $a$).
   • Traverse the edges following the rules of Fleury's algorithm.

3. Traversal (one possible Euler circuit):

   • Starting at $a$: $a \to b \to c \to h \to g \to f \to e \to d \to b \to d \to c \to a$

Result:

The Euler circuit for the graph is: $a \to b \to c \to h \to g \to f \to e \to d \to b \to d \to c \to a$

Let me know if you would like a detailed diagram or further clarifications.

5 Related Questions:

1. What is the difference between Euler circuits and Euler paths?
2. How does Fleury’s algorithm handle disconnected graphs?
3. Can Fleury’s algorithm work for graphs with odd-degree vertices? Why or why not?
4. How does removing a "bridge" edge impact the graph's connectivity?
5. How can we determine the starting point of an Euler circuit if multiple options exist?

Tip: When solving Euler circuit problems, always verify vertex degrees and connectivity first to ensure the graph qualifies!