This paper considers inclusion relations between the following four classes of real square matrices:
The diagonally stable matrices ${\cal A}$ consisting of those matrices A such that $AD+DA\sp T$ is positive definite for some positive definite diagonal matrix D, the positive stable matrices ${\cal L}$ consisting of those matrices A such that $AX+XA\sp T$ is positive definite for some positive definite matrix X, the P-matrices ${\cal P}$ consisting of those A all of whose principal minors are positive, and the semi-positive matrices ${\cal S}$ consisting of all A such that $Ax>0$ for some $x>0.$
In particular, it is shown that if A is a P-matrix and if the nondirected graph of A is a forest, then A is diagonally stable. (The nondirected graph of an $n\times n$ matrix A is the graph whose vertices are 1,2,..., and whose edges are the pairs $\{$ i,$j\}$, $i\ne j$, for which either $a\sb{ij}$ or $a\sb{ji}$ is nonzero.) This result, in view of previously known results, yields for those matrices whose nondirected graph is a forest the relations ${\cal L} \Leftarrow {\cal A} \Leftrightarrow {\cal P} \Rightarrow {\cal S}$ where $\Rightarrow$ means ”is contained in” and the absence of an implication implies a counterexample. Partial results and some open questions are given for real spectra matrices and for a class of matrices called $\omega$ -matrices.