arXiv 1702.042452017-02-14math.CO

关于图的块数

On the block number of graphs

Daniel Weißauer

摘要：

图$G$中的一个$k$ -块是至少由$k$顶点组成的极大集，其中没有两个顶点可以通过删除少于$k$的顶点而在$G$中分离。$G$的块号$β(G)$是$G$包含$k$块的最大整数$k$。我们证明了一个结构定理的图没有$(k+1)$块，表明每一个这样的图有树分解，其中每一个躯干最多$k$度$2k^2$或更大的顶点。这产生了一个定性的对偶，因为每个允许这样的分解的图最多有$2k^2$块数。我们还研究了图类$\mathcal{G}$中的$k$ -块，并证明了对于某些常数$c = c( \mathcal{G})$，每个$G \in \mathcal{G}$都满足$β(G) \leq c\sqrt[3]{|G|}$.此外，我们还证明了树宽至少为$2k^2$的图都有一个包含$k$ -块的子图。这个界限是最好的可能性，直到一个乘法常数。

英文摘要：

A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by deleting fewer than $k$ vertices. The block number $β(G)$ of $G$ is the maximum integer $k$ for which $G$ contains a $k$-block. We prove a structure theorem for graphs without a $(k+1)$-block, showing that every such graph has a tree-decomposition in which every torso has at most $k$ vertices of degree $2k^2$ or greater. This yields a qualitative duality, since every graph that admits such a decomposition has block number at most $2k^2$. We also study $k$-blocks in graphs from classes of graphs $\mathcal{G}$ that exclude some fixed graph as a topological minor, and prove that every $G \in \mathcal{G}$ satisfies $β(G) \leq c\sqrt[3]{|G|}$ for some constant $c = c( \mathcal{G})$. Moreover, we show that every graph of tree-width at least $2k^2$ has a minor containing a $k$-block. This bound is best possible up to a multiplicative constant.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文