Fixed points and graph images for minimal maps on the once punctured torus

A fixed point detection theorem for a family of maps defined on the once punctured torus is proved. As a consequence, we produce an example of a homotopy class [f] of self-maps on the once punctured torus that illustrates the following: (i) there is a map in the homotopy class that has no fixed points, and (ii) if the image of f lies in a 1-complex that embeds as a homotopy equivalence, then f must have a fixed point.