On the tree cover number and the positive semidefinite maximum nullity of a graph

For a simple graph $G=(V,E),$ let $\mathcal{S}_+(G)$ denote the set of real positive semidefinite matrices $A=(a_{ij})$ such that $a_{ij}\neq 0$ if $\{i,j\}\in E$ and $a_{ij}=0$ if $\{i,j\}\notin E$. The maximum positive semidefinite nullity of $G$, denoted ${\rm M}_+(G),$ is $\max\{{\rm null}(A)|A\in \mathcal{S}_+(G)\}.$ A tree cover of $G$ is a collection of vertex-disjoint simple trees occurring as induced subgraphs of $G$ that cover all the vertices of $G$. The tree cover number of $G$, denoted $T(G)$, is the cardinality of a minimum tree cover. It is known that the tree cover number of a graph and the maximum positive semidefinite nullity of a graph are equal for outerplanar graphs, and it was conjectured in 2011 that $T(G)\leq {\rm M}_+(G)$ for all graphs [Barioli et al., Minimum semidefinite rank of outerplanar graphs and the tree cover number, {\em Elec. J. Lin. Alg.,} 2011]. We show that the conjecture is true for certain graph families. Furthermore, we prove bounds on $T(G)$ to show that if $G$ is a connected outerplanar graph on $n\geq 2$ vertices, then ${\rm M}_+(G)=T(G)\leq \left\lceil\frac{n}{2}\right\rceil$, and if $G$ is a connected outerplanar graph on $n\geq 6$ vertices with no three or four cycle, then ${\rm M}_+(G)=T(G)\leq \frac{n}{3}$. We characterize connected outerplanar graphs with ${\rm M}_+(G)=T(G)=\left\lceil\frac{n}{2}\right\rceil,$ and for each cactus graph $G$, we give a formula for computing $T(G)$ (and therefore ${\rm M}_+(G)$)).