The curve graph of a translation surface is a graph whose vertices are homotopy classes of regular closed geodesics. Two vertices are joined by an edge if the corresponding geodesics are disjoint. In general, this graph is not connected. In this talk, we introduce an extension of the curve graph that is connected. We show that the extended curve graphs of hyperelliptic translation surfaces share many of the same properties as the classical curve graphs of surfaces. We also give a characterization of hyperelliptic Veech surfaces in terms of their extended curve graphs.