去除诱导爪和菱形的多项式核化
Polynomial kernelization for removing induced claws and diamonds
摘要：
一个图称为（claw，diamond）-free，如果它既不包含一个claw（一个$K_{1,3}$）也不包含一个diamond（一个去掉边的$K_4$）作为导出子图.无等距（爪形，菱形）图可以被表征为无三角形图的线图，或线性多米诺骨牌，即，图中的每一个顶点是在最多两个最大团和每一个边缘是在一个最大团。本文研究了无（claw，diamond）边删除问题的参数化复杂性，其中给定一个图$G$和一个参数$k$，问题是能否从$G$中最多删除$k$边以得到一个无（claw，diamond）图.我们的主要结果是，这个问题承认一个多项式核。我们补充这一发现证明，即使在最大程度$6$的情况下，问题是NP完全的，不能在时间$2^{o(k)}\cdot |V(G)|^{O(1)}$解决，除非指数时间假设失败
英文摘要：
A graph is called (claw,diamond)-free if it contains neither a claw (a $K_{1,3}$) nor a diamond (a $K_4$ with an edge removed) as an induced subgraph. Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex is in at most two maximal cliques and every edge is in exactly one maximal clique. In this paper we consider the parameterized complexity of the (claw,diamond)-free Edge Deletion problem, where given a graph $G$ and a parameter $k$, the question is whether one can remove at most $k$ edges from $G$ to obtain a (claw,diamond)-free graph. Our main result is that this problem admits a polynomial kernel. We complement this finding by proving that, even on instances with maximum degree $6$, the problem is NP-complete and cannot be solved in time $2^{o(k)}\cdot |V(G)|^{O(1)}$ unless the Exponential Time Hypothesis fail
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。
