The singularity degree of a semidefinite programming problem is the smallest number of facial reduction steps to make the problem strictly feasible. We introduce two new graph parameters, called the singularity degree and the nondegenerate singularity degree, based on the singularity degree of the positive semidefinite matrix completion problem. We give a characterization of the class of graphs whose parameter value is equal to one for each parameter. Specifically, we show that the singularity degree of a graph is equal to one if and only if the graph is chordal, and the nondegenerate singularity degree of a graph is equal to one if and only if the graph is the clique sum of chordal graphs and $K_4$-minor free graphs. We also show that the singularity degree is bounded by two if the treewidth is bounded by two, and exhibit a family of graphs with treewidth three, whose singularity degree grows linearly in the number of vertices.