Determine whether the graph has an Euler path, an Euler circuit, or neither. If

ID: 3141810 • Letter: D

Question

Determine whether the graph has an Euler path, an Euler circuit, or neither.

If the graph has an Euler path or circuit, use trial and error or Fleury's algorithm to find one.

1. Choose the correct answer below. The graph has

a. The graph has an Euler path (but not an Euler circuit).

b. The graph has an Euler circuit.

c. The graph has neither an Euler path nor an Euler circuit.

2. If the graph has an Euler path or circuit, use trial and error or Fleury's Algorithm to find one starting at I.

Choose the correct answer below.

a. I, H, D, C, A , B, C, D, E, G, F, D

b. I, H, D, E, G, F, D, C, B, A, C

c. I, D, F, G, E, D, C, B, A, C

d. There is not Euler path or circuit.

H IGF DE

1. If a graph has to have a Euler's circuit, all vertices must have even degree. But C and I have degree 3.

So there is no Euler cycle.

A singly connected graph has an Euler trail if and only if exactly zero or two vertices have odd degree.

Since only C and I have degree 3, there is an Euler path.

So the answer is a. The graph has an Euler path (but not an Euler circuit).

2. We notice that H and I are only connected via D.

So we start with HI followed by DH to reach D.

Next we notice that CD forms a bridge with ABC.

So we visit DE first, followed by EG, FG,DF.

We then visit CD, BC, AB and AC.

Thus the Euler path is I, H, D, E, G, F, D, C, B, A,C.

So the answer is c.

(Note: Other possibilities are I, H, D, E, G, F, D, C, A, B,C; I, H, D, F, G, E, D, C, B, A, C and I, H, D, F, G, E, D, C, A, B, C)

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