Solving problems on generalized convex graphs via mim-width

A bipartite graph $G=(A,B,E)$ is $H$-convex for some family of graphs $H$ if there exists a graph $H\in H$ with $V\left(H\right)=A$ such that the neighbours in A of each $b\in B$ induce a connected subgraph of H. Many $\mathrm{NP}$-complete problems are polynomial-time solvable for $H$-convex graphs when $H$ is the set of paths. The underlying reason is that the class has bounded mim-width. We extend this result to families of $H$-convex graphs where $H$ is the set of cycles, or $H$ is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least 3. As a consequence, we strengthen many known results via one general and short proof. We also show that the mim-width of $H$-convex graphs is unbounded if $H$ is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least 3.