Treewidth of Erdös-Rényi Random Graphs, Random Intersection Graphs, and Scale-Free Random Graphs

Large-Treewidth Graph Decompositions and Applications

Treewidth is a graph parameter that plays a fundamental role in several structural and algorithmic results. We study the problem of decomposing a given graph $G$ into node-disjoint subgraphs, where each subgraph has sufficiently large treewidth. We prove two theorems on the tradeoff between the number of the desired subgraphs $h$, and the desired lower bound $r$ on the treewidth of each subgraph. The theorems assert that, given a graph $G$ with treewidth $k$, a decomposition with parameters $h,r$ is feasible whenever $hr^2 \le k/\polylog(k)$, or $h^3r \le k/\polylog(k)$ holds. We then show a f…

Polynomial Gap Extensions of the Erdős-Pósa Theorem

Given a graph $H$, we denote by ${\cal M}(H)$ all graphs that can be contracted to $H$. The following extension of the Erdős-Pósa Theorem holds: for every $h$-vertex planar graph $H$, there exists a function $f_{H}$ such that every graph $G$, either contains $k$ disjoint copies of graphs in ${\cal M}(H)$, or contains a set of $f_{H}(k)$ vertices meeting every subgraph of $G$ that belongs in ${\cal M}(H)$. In this paper we prove that this is the case for every graph $H$ of pathwidth at most 2 and, in particular, that $f_{H}(k) = 2^{O(h^2)}\cdot k^{2}\cdot \log k$. As a main ingredient of the pr…

四分之一积分与二分之一积分的有向环装填

Packing Directed Cycles Quarter- and Half-Integrally

本研究针对有向图的环装填与反馈顶点集问题，证明了四分之一积分和二分之一积分环装填数下反馈顶点集大小的多项式界，并得到高有向树宽子图四分之一积分装填的一般性结果。