In this work, we investigate the algorithmic aspects of two natural extensions of hereditary classes: the edge-apex class and the edge-add class, recently introduced by Singh and Sivaraman. These are defined as the graph classes obtained by at most one edge deletion or one non-edge addition, respectively, from a hereditary class $\mathcal{G}$. Building on earlier results showing that both classes remain hereditary and admit finite forbidden induced subgraph characterizations whenever $\mathcal{G}$ does, we focus on the Weighted Maximum Clique Problem (WMCP) and the Weighted Maximum Independent Set Problem (WMISP). We first present algorithms for WMCP and WMISP on both the edge-apex and edge-add classes of hereditary graph classes. Extending this framework, we introduce the notion of the $\mathcal{G}$-edge distance of a graph $G$, denoted by $ξ_{\mathcal{G}}(G)$, which quantifies how far $G$ is from the class $\mathcal{G}$ in terms of the minimum number of edge deletions or non-edge additions needed to transform it into a member of $\mathcal{G}$. By parameterizing with respect to this distance, we show that both WMCP and WMISP can be solved in $O^*(2^k)$ time on graphs whose $\mathcal{G}$-edge distance is $k$, provided these problems admit polynomial-time algorithms within the class $\mathcal{G}$. This result extends earlier algorithmic characterizations of the single edge-apex and edge-add classes to the more general setting of $k$-edge-distant graphs. By combining our general results with known properties of transitive graphs, we show that WMCP and WMISP can be solved in $O^*(2^k)$ time for graphs with transitive-edge distance $k$.

翻译：本文研究两类自然遗传图类扩展的算法性质：即由Singh与Sivaraman近期引入的边缘顶点类与边缘添加类。这两类图分别定义为从遗传类$\mathcal{G}$中通过至多一次边删除或一次非边添加操作所获得的图类。基于前期结果（表明只要$\mathcal{G}$具有有限禁止诱导子图刻画，则这两类图仍保持遗传性且同样具有有限禁止诱导子图刻画），我们重点研究加权最大团问题（WMCP）与加权最大独立集问题（WMISP）。首先，我们针对遗传图类的边缘顶点类与边缘添加类，分别给出WMCP与WMISP的算法。在此基础上，我们引入图$G$的$\mathcal{G}$-边距离概念（记为$ξ_{\mathcal{G}}(G)$），该距离量化了$G$与类$\mathcal{G}$的偏离程度，具体定义为将$G\)转换为$\mathcal{G}$中成员所需的最小边删除或非边添加操作次数。通过以该距离为参数化变量，我们证明：若WMCP与WMISP在$\mathcal{G}$类内存在多项式时间算法，则对$\mathcal{G}$-边距离为$k$的图，这两个问题可在$O^*(2^k)$时间内求解。该结果将此前针对单一边缘顶点类与边缘添加类的算法刻画推广至更一般的$k$-边距离图类。最后，结合通用结果与传递图类的已知性质，我们证明：对传递边距离为$k$的图，WMCP与WMISP可在$O^*(2^k)$时间内求解。