A bipartite graph $G=(A,B,E)$ is ${\cal H}$-convex, for some family of graphs ${\cal H}$, if there exists a graph $H\in {\cal H}$ with $V(H)=A$ such that the set of neighbours in $A$ of each $b\in B$ induces a connected subgraph of $H$. A variety of well-known $\mathsf{NP}$-complete problems, including \textsc{Dominating Set}, \textsc{Feedback Vertex Set}, \textsc{Induced Matching} and \textsc{List $k$-Colouring}, become polynomial-time solvable for ${\mathcal H}$-convex graphs when ${\mathcal H}$ is the set of paths. In this case, the class of ${\mathcal H}$-convex graphs is known as the class of convex graphs. The underlying reason is that the class of convex graphs has bounded mim-width. We extend the latter result to families of ${\mathcal H}$-convex graphs where (i) ${\mathcal H}$ is the set of cycles, or (ii) ${\mathcal H}$ is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least $3$. As a consequence, we can re-prove and strengthen a large number of results on generalized convex graphs known in the literature. To complement result (ii), we 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 arbitrarily large number of vertices of degree at least $3$. In this way we are able to determine complexity dichotomies for the aforementioned graph problems.

中文翻译：

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通过mim-width解决广义凸图问题

二部图 $G=(A,B,E)$ 是 ${\cal H}$-凸的，对于一些图族 ${\cal H}$，如果存在图 $H\in {\cal H}$ 与 $V(H)=A$ 使得每个 $b\in B$ 的 $A$ 中的邻居集合归纳出 $H$ 的连通子图。各种著名的$\mathsf{NP}$-完全问题，包括\textsc{Domination Set}、\textsc{Feedback Vertex Set}、\textsc{Induced Matching} 和\textsc{List $k$-Colouring} ，当 ${\mathcal H}$ 是路径集时，成为 ${\mathcal H}$-凸图的多项式时间可解。在这种情况下，${\mathcal H}$-凸图类被称为凸图类。根本原因是凸图类具有有界的mim-width。我们将后一个结果扩展到 ${\mathcal H}$-凸图系列，其中 (i) ${\mathcal H}$ 是循环的集合，或 (ii) ${\mathcal H}$ 是最大度数有界且度数至少为 $3$ 的顶点数有界的树的集合。因此，我们可以在文献中已知的广义凸图上重新证明和加强大量结果。为了补充结果（ii），我们表明，如果 ${\mathcal H}$ 是具有任意大最大度数或任意大数量的树的集合，则 ${\mathcal H}$-凸图的 mim-width 是无界的。度数的顶点至少 $3$。通过这种方式，我们能够确定上述图问题的复杂性二分法。我们表明，如果 ${\mathcal H}$ 是具有任意大的最大度数或任意大量度数至少为 $3$ 的顶点的树的集合，则 ${\mathcal H}$-凸图的 mim-width 是无界的. 通过这种方式，我们能够确定上述图问题的复杂性二分法。我们表明，如果 ${\mathcal H}$ 是具有任意大的最大度数或任意大量度数至少为 $3$ 的顶点的树的集合，则 ${\mathcal H}$-凸图的 mim-width 是无界的. 通过这种方式，我们能够确定上述图问题的复杂性二分法。