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 is open. The complexities of recognising many subclasses of SOGs are known. Weconsider 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

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 is open. The complexities of recognising many subclasses of SOGs are known. Weconsider 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

The vertex leafage of chordal graphs

Every chordal graph $G$ can be represented as the intersection graph of a collection of subtrees of a host tree, a so-called {\em tree model} of $G$. The leafage $\ell(G)$ of a connected chordal graph $G$ is the minimum number of leaves of the host tree of a tree model of $G$. The vertex leafage \vl(G) is the smallest number $k$ such that there exists a tree model of $G$ in which every subtree has at most $k$ leaves. The leafage is a polynomially computable parameter by the result of \cite{esa}. In this contribution, we study the vertex leafage. We prove for every fixed $k\geq 3$ that deciding whether the vertex leafage of a given chordal graph is at most $k$ is NP-complete by proving a stronger result, namely that the problem is NP-complete on split graphs with vertex leafage of at most $k+1$. On the other hand, for chordal graphs of leafage at most $\ell$, we show that the vertex leafage can be calculated in time $n^{O(\ell)}$. Finally, we prove that there exists a tree model that realizes both the leafage and the vertex leafage of $G$. Notably, for every path graph $G$, there exists a path model with $\ell(G)$ leaves in the host tree and it can be computed in $O(n^3)$ time