通过Mim-Width解决广义凸图上的问题
Solving Problems on Generalized Convex Graphs via Mim-Width
AI总结：
该研究通过mim-width界定广义凸图（基于环或有界树）的宽度，证明相关NP完全问题可多项式求解，并给出复杂性二分法。
摘要：
一个二分图$G=(A,B,E)$是${\cal H}$-凸的，对于某个图族${\cal H}$，如果存在一个图$H\in {\cal H}$，其中$V(H)=A$，使得每个$b\in B$在$A$中的邻居集合在$H$中诱导出一个连通子图。许多$\mathsf{NP}$-完全问题，包括支配集、反馈顶点集、诱导匹配和列表$k$-着色等问题，当${\mathcal H}$是路径集合时，对于${\mathcal H}$-凸图可以在多项式时间内解决。在这种情况下，${\mathcal H}$-凸图的类被称为凸图类。根本原因是凸图类具有有界的mim-width。我们将这一结果扩展到${\mathcal H}$-凸图的族，其中(i)${\mathcal H}$是环的集合，或(ii)${\mathcal H}$是最大度有界且度数至少为3的顶点数量有界的树的集合。因此，我们可以重新证明并加强文献中已知的关于广义凸图的大量结果。为了补充结果(ii)，我们证明如果${\mathcal H}$是最大度任意大或度数至少为3的顶点数量任意大的树的集合，则${\mathcal H}$-凸图的mim-width是无界的。通过这种方式，我们能够为上述图问题确定复杂性二分法。随后，我们进行了更精细的宽度参数分析，这更清楚地显示了对于${\cal H}$-凸图的类，哪些宽度参数是有界的。
英文摘要：
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 $H$. Many $\mathsf{NP}$-complete problems, including problems such as Dominating Set, Feedback Vertex Set, Induced Matching and 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 an 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. Afterwards we perform a more refined width-parameter analysis, which shows even more clearly which width parameters are bounded for classes of ${\cal H}$-convex graphs.
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