Invitation to torso parameters of annotated graphs

Overview & Synopsis

An annotated graph is a pair $(G,S)$, where $G$ is a graph and $S$ is a subset of vertices of $G$. We present a unified approach for definitions of annotated versions of classical graph parameters such as treewidth, pathwidth, treedepth, etc. A generic way to incorporate a distinguished set of vertices into a graph parameter is via torsos. Given a graph $G$ and a subset $S'\subseteq V(G)$, we define $\mathrm{torso}_G(S')$ to be the graph with vertex set $S'$ in which distinct vertices $u,v\in S'$ are adjacent if and only if there exists a $u$-$v$ path in $G$ whose internal vertices lie outside $S'$. Now let $\mathrm{p}$ be any graph parameter. For an annotated graph $(G,S)$, we define the torso $\mathrm{p}$ of $(G,S)$ by \[\mathrm{p}(G,S) = \min\{\mathrm{p}(\mathrm{torso}_G(S')) \mid S \subset S' \subset V(G)\}.\] For a minor-closed class of annotated graphs $\mathcal{C}$, we discuss equivalent conditions for $\mathrm{p}(G,S)$ bounded for all $(G,S)\in\mathcal{C}$ in terms of annotated graphs excluded in $\mathcal{C}$. In many cases, we collect known results from the literature. We include two new arguments for $\mathrm{p}$ being the $2$-treedepth and the feedback vertex number. Among other statements, we also include a variant of the Erdős-Pósa property for annotated cycles. We propose some new research directions.