AI 中文总结
研究不含\(k\)个两两不相邻平面图\(H\)子图模型的图\(G\)，通过证明稀疏\(kH -\)自由图有相关性质，得出可移除\(f(k,H)\)个邻域使其\(H -\)子图自由，并给出\(kH -\)自由图最大独立集问题的拟多项式算法。
AI 中文摘要
埃尔德什 - 波萨定理表明，每个不含\(k\)个不相交圈的图\(G\)包含一个\(f(k)\)个顶点的集合\(X\)，使得\(G\setminus X\)无圈。罗伯逊和西摩证明该性质对任何平面图\(H\)的\(H -\)子图模型也成立。我们证明这种粗粒度图论观点推广到距离至少为\(2\)与半径为\(1\)的球的情况，得到平面子图的诱导埃尔德什 - 波萨性质。即不含\(k\)个两两不相邻的平面图\(H\)子图模型的图\(G\)，通过移除\(f(k,H)\)个邻域可变为\(H -\)子图自由。证明依赖于稀疏\(kH -\)自由图有线性数量的独立大突出部分这一事实。同一方法表明稀疏\(kH -\)自由图通过删除\(O(\log n)\)个顶点可变为\(H -\)子图自由，从而给出了\(kH -\)自由图的最大独立集问题的拟多项式算法。
英文摘要
The Erdős--Pósa theorem asserts that every graph $G$ with no $k$ disjoint cycles contains a set $X$ of $f(k)$ vertices such that $G\setminus X$ has no cycle. Robertson and Seymour showed that this Erdős--Pósa property also holds for $H$-minor models of any planar graph $H$. Equivalently, if $G$ has no $k$ minor models of $H$ pairwise at distance at least 1 (i.e. disjoint), then one can remove $f(k,H)$ balls of radius 0 (i.e. vertices) to make the graph $H$-minor free. We show that this coarse graph theory point of view generalizes to distance at least 2 versus radius 1 balls, yielding the induced Erdős--Pósa property for planar minors. Namely, every graph $G$ which does not contain $k$ pairwise non-adjacent minor models of a planar graph $H$ (we say that $G$ is $kH$-free) can be made $H$-minor free by removing $f(k,H)$ neighborhoods. The proof relies on the fact that sparse $kH$-free graphs have linearly many independent large protrusions. The same method gives that sparse $kH$-free graphs can be made $H$-minor free by deleting $O(\log n)$ vertices (and thus have logarithmic tree-width). This gives a quasi-polynomial algorithm for the Maximum Independent Set problem for $kH$-free graphs.
Comments21 pages
