Recognising the overlap graphs of subtrees of restricted trees is hard

The overlap graphs of subtrees in a tree (SOGs) generalise many other graphs classes with set representation characterisations. The complexity of recognising SOGs in open. The complexities of recognising many subclasses of SOGs are known. We consider several subclasses of SOGs by restricting the underlying tree. For a fixed integer $k \geq 3$, we consider: \begin{my_itemize} \item The overlap graphs of subtrees in a tree where that tree has $k$ leaves \item The overlap graphs of subtrees in trees that can be derived from a given input tree by subdivision and have at least 3 leaves \item The overlap and intersection graphs of paths in a tree where that tree has maximum degree $k$ \end{my_itemize} We show that the recognition problems of these classes are NP-complete. For all other parameters we get circle graphs, well known to be polynomially recognizable.