发表机构
Departamento de Computación, Universidad de Buenos Aires; CONICET（布宜诺斯艾利斯大学计算机科学系; 阿根廷国家科学研究委员会）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对 $K_r-e$-自由图三明治问题，提出一种对树状度敏感的确定性算法，将时间复杂度从 $O(n^r m_2)$ 改进到 $O(n+\alpha(G_2)^{r-3}m_2)$，并应用于探针识别问题，获得更优时间界。
AI 中文摘要
对于固定的整数 $r\geq4$，$K_r-e$-自由图三明治问题询问：给定同一顶点集上的图 $G_1\subseteq G_2$，是否存在一个介于两者之间的诱导-$K_r-e$-自由图 $H$。我们给出一个确定性算法，其时间和空间复杂度为 $O(n+\alpha(G_2)^{r-3}m_2)$，其中 $m_2=|E(G_2)|$，$\alpha(G_2)$ 是 $G_2$ 的树状度（arboricity）。特别地，无菱形（diamond-free）情形的时间复杂度为 $O(n+\alpha(G_2)m_2)$。这改进了先前已知的强制边闭包（forced-edge closure）的直接 $O(n^r m_2)$ 实现。我们的实现维护由 $(r-3)$-团索引的公共邻域组件。一个过滤前沿（filtered frontier）支持在团列举界内进行合并，而完成事件避免重复搜索受影响的团。在可行实例上，输出包含于每个可行三明治中，与处理顺序无关。将闭包应用于 $(G,K_n)$ 可得到分区和非分区探针 $K_r-e$-自由识别问题的 $O(n^{r-1})$ 时间界，改进了由直接三明治闭包得到的 $O(n^{r+2})$ 界。我们还描述了一个基于相同局部特征的直接静态识别器。
英文摘要
For a fixed integer $r\geq4$, the $K_r-e$-free graph sandwich problem asks whether, given graphs $G_1\subseteq G_2$ on the same vertex set, there is an induced-$K_r-e$-free graph $H$ between them. We give a deterministic algorithm taking $O(n+α(G_2)^{r-3}m_2)$ time and space, where $m_2=|E(G_2)|$ and $α(G_2)$ is the arboricity of $G_2$. In particular, the diamond-free case takes $O(n+α(G_2)m_2)$ time. This improves the direct $O(n^r m_2)$ implementation of the previously known forced-edge closure. Our implementation maintains components of common neighborhoods indexed by $(r-3)$-cliques. A filtered frontier supports their merges within the clique-listing bound, while completion events avoid repeatedly searching for affected cliques. On feasible instances the output is contained in every feasible sandwich, independently of processing order. Applying the closure to $(G,K_n)$ gives an $O(n^{r-1})$-time bound for partitioned and nonpartitioned probe $K_r-e$-free recognition, improving the $O(n^{r+2})$ bound obtained from the direct sandwich closure. We also describe a direct static recognizer based on the same local characterization.
Comments10 pages. Expanded implementation details and complexity justification; clarified the word-RAM model, clique-event processing, and the discussion of previous probe recognition algorithms. No change to the main results
