Solving problems on generalized convex graphs via mim-width

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