Abstract
Given a finite set of graphs ${\cal H}$ and a non-negative integer $k$, we define ${\cal A}_{k}({\cal H})$ as the set containing every graph $G$ that has $k$ vertices whose removal provides a graph without any of the graphs in ${\cal H}$ as a minor.It is known that if ${\cal H}$ contains at least one planar graph then each obstruction in ${\cal A}_{k}({\cal H})$ has at most $k^{c_{\cal H}}$ vertices, for some $c_{\cal H}$ depending only on the choice of ${\cal H}$. In this paper, we investigate the size of the graphs in ${\cal A}_{k}({\cal H})$ that belong to certain classes of sparse graphs. In particular, we prove that for every graph $F$, if ${\cal H}$ contains at least one planar graph and only connected graphs, all graphs in ${\cal A}_{k}({\cal H})$ that are $F$-topological minor-free have at most $c_{F,{\cal H}}\cdot k$ vertices, where $c_{F,{\cal H}}$ depends exclusively on the choice of ${\cal H}$ and $F$.Our result is a consequence of two more general conditions on graph parameters, namely the Finite Integer Index Property and Protrusion Decomposability, that can serve as a general framework for proving linear bounds for obstructions.