Being even slightly shallow makes life hard

We study the computational complexity of identifying dense substructures, namely $r\mathrm{/}2$-shallow topological minors and $r$-subdivisions. Of particular interest is the case when $r=1$, when these substructures correspond to very localized relaxations of subgraphs. Since Densest Subgraph can be solved in polynomial time, we ask whether these slight relaxations also admit efficient algorithms. In the following, we provide a negative answer: Dense $r\mathrm{/}2$-Shallow Topological Minor and Dense $r$-Subdivsion are already NP-hard for $r=1$ in very sparse graphs. Further, they do not admit algorithms with running time $2^{o({\mathbf{t}\mathbf{w}}^2)}{n}^{O(1)}$ when parameterized by the treewidth of the input graph for $r\ge 2$ unless ETH fails.