用于无钻石边缘删除等的三次顶点核
A cubic vertex kernel for Diamond-free Edge Deletion and more
摘要：
菱形是通过从四个顶点上的完全图中移除一条边而获得的图。一个图是无菱形的，如果它不包含诱导菱形。无钻石边删除问题询问在输入图$G$中是否存在最多$k$条边，其删除导致无钻石图。对于这个问题，Fellows等人找到了一个由$O(k^4$）个顶点组成的多项式核.（Discrete Optimization，2011）.本文给出了一种改进的无钻石边删除的$O(k^3)$顶点核。进一步，我们给出了一个相关的无{Diamond，K_t}边删除问题的$O(k^2)$顶点核，其中$t\geq 4$为任意固定整数.为了补充我们的结果，我们证明了这些问题是NP-完全的，即使是$K_4$ -free图，也不能在亚指数时间内解决（即，$2^{o(|G|)}$）或参数化的亚指数时间（即，$2^{o(k)}\cdot |G|^{O(1)}$），除非指数时间假设失败。我们的减少意味着硬度和下界的一般类的问题，这些问题作为一个特殊的情况。
英文摘要：
A diamond is a graph obtained by removing an edge from a complete graph on four vertices. A graph is diamond-free if it does not contain an induced diamond. The Diamond-free Edge Deletion problem asks whether there exist at most $k$ edges in the input graph $G$ whose deletion results in a diamond-free graph. For this problem, a polynomial kernel of $O(k^4$) vertices was found by Fellows et. al. (Discrete Optimization, 2011). In this paper, we give an improved kernel of $O(k^3)$ vertices for Diamond-free Edge Deletion. Further, we give an $O(k^2)$ vertex kernel for a related problem {Diamond,K_t}-free Edge Deletion, where $t\geq 4$ is any fixed integer. To complement our results, we prove that these problems are NP-complete even for $K_4$-free graphs and can be solved neither in subexponential time (i.e., $2^{o(|G|)}$) nor in parameterized subexponential time (i.e., $2^{o(k)}\cdot |G|^{O(1)}$), unless Exponential Time Hypothesis fails. Our reduction implies the hardness and lower bound for a general class of problems, where these problems come as a special case.
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