38. An Euler circuit visits a four-valent vertex X, such as the one in the accompanying graph, by using the edges AX and XB consecutively, and then using CX and XD consecutively. When this happens, we say that the Euler circuit cuts through at X.

Suppose G is a four-valent graph such as that in the diagram below. Is it possible to find an Euler circuit of this graph that never cuts through any vertex? Explain why it might be desirable to find an Euler circuit of this special kind in an applied situation.