For the pants, yes. In general, no. To prove this for the pants, classify all geodesic arcs and just observe the result. There are many ways to find a "no" example in the general case; the first one that came to my mind was taking a double cover.
EDIT - I see that this is a near-duplicate of a closed question. You could improve your question by giving some motivation. Reading the FAQ will be very useful in writing questions that get good answers. In particular please see http://mathoverflow.net/faq#whatnot
