图问题核化算法中的$\mathbf{c}$ -闭包
Exploiting $\mathbf{c}$-Closure in Kernelization Algorithms for Graph Problems
摘要：
一个图是c-闭的，如果每对顶点至少有c个公共邻居是相邻的。图G的c-闭包是使G是c-闭的最小数。Fox等人[ICALP '18]定义了c-闭包，并在团枚举的背景下对其进行了研究。我们表明c-闭包可以应用于几个经典图问题的核化算法中。证明了支配集允许一个k^O（c）的核，诱导匹配允许一个O（c^7*k^8）顶点的核，无冗余集允许一个O（c^5/2）*k^3）顶点的核.我们的核化利用了c-闭图具有多项式有界Ramsey数的事实，正如我们所展示的。
英文摘要：
A graph is c-closed if every pair of vertices with at least c common neighbors is adjacent. The c-closure of a graph G is the smallest number such that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated it in the context of clique enumeration. We show that c-closure can be applied in kernelization algorithms for several classic graph problems. We show that Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show.
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