The class doughnut graphs is a subclass of 5-connected planar graphs. In a planar embedding of a doughnut graph of n vertices there are two vertex-disjoint faces each having exactly n/4 vertices and each of all the other faces has exactly three vertices. Recently the class of doughnut graphs is introduced to show that a graph in this class admits a straight-line grid drawing with linear area and hence any spanning subgraph of a doughnut graph also admits a straight-line grid drawing with linear area. But recognition of a spanning subgraph of a doughnut graph is a non-trivial problem, since recognition of a spanning subgraph of a given graph is an NP-complete problem in general. In this paper, we establish a necessary and sufficient condition for a 4-connected planar graph G to be a spanning subgraph of a doughnut graph. We also give a linear-time algorithm to augment a 4-connected planar graph G to a doughnut graph if G satisfies the necessary and sufficient condition.