  Let G be an undirected graph on n vertices and let S(G) be the set of all
real symmetric n x n matrices whose nonzero off-diagonal entries occur in
exactly the positions corresponding to the edges of G. The inverse inertia
problem for G asks which inertias can be attained by a matrix in S(G). We give
a complete answer to this question for trees in terms of a new family of graph
parameters, the maximal disconnection numbers of a graph. We also give a
formula for the inertia set of a graph with a cut vertex in terms of inertia
sets of proper subgraphs. Finally, we give an example of a graph that is not
inertia-balanced, and investigate restrictions on the inertia set of any graph.
