A graph is a $(k,\ell)$-graph if its vertex set can be partitioned into $k$ independent sets and $\ell$ cliques. This family simultaneously generalizes split, bipartite, and co-bipartite graphs. While the recognition problem is NP-complete whenever $k\geq 3$ or $\ell\geq 3$, the remaining small cases are polynomial-time solvable. In this paper we revisit the known recognition algorithms for the first nontrivial polynomial cases, namely $(2,1)$-, $(1,2)$-, and $(2,2)$-graphs, and show how to remove specific bottlenecks in their existing recognition procedures. For $(2,1)$-graphs, we show that the extra quadratic enumeration in the algorithm of Brandstädt, Le and Szymczak can be avoided by exploiting the structure of a shortest odd cycle in the relevant residual graph, reducing the running time from $O((n+m)^2)$ to $O(n(n+m))$. By complementation, this yields an $O(n(n+\overline m))$-time recognition algorithm for $(1,2)$-graphs, where $\overline m$ denotes the number of edges of the complement graph. For $(2,2)$-graphs, we refine the sparse-dense partition framework of Feder, Hell, Klein and Motwani by restricting the local-search enumeration to sets that are simultaneously bipartite and co-bipartite, and by using the improved algorithms for $(2,1)$- and $(1,2)$-graphs as preprocessing tools. This gives an $O(n^4(n+\min\{m,\overline m\})^3)$-time recognition algorithm for $(2,2)$-graphs.

翻译：如果图的顶点集可划分为$k$个独立集和$\ell$个团，则称该图为$(k,\ell)$-图。该图族同时推广了分裂图、二部图和补二部图。虽然当$k\geq 3$或$\ell\geq 3$时识别问题属于NP完全问题，但其余小规模情形可在多项式时间内求解。本文重新审视了首批非平凡多项式情形（即$(2,1)$-图、$(1,2)$-图和$(2,2)$-图）的已知识别算法，并展示了如何消除其现有识别流程中的特定瓶颈。对于$(2,1)$-图，我们证明通过利用相关残差图中最短奇环的结构，可避免Brandstädt、Le和Szymczak算法中额外的二次枚举，从而将运行时间从$O((n+m)^2)$降至$O(n(n+m))$。通过补图对称性，该结果为$(1,2)$-图提供了$O(n(n+\overline m))$时间的识别算法，其中$\overline m$表示补图的边数。对于$(2,2)$-图，我们改进了Feder、Hell、Klein和Motwani的稀疏-稠密划分框架：将局部搜索枚举限制在同时为二部图和补二部图的集合上，并利用改进后的$(2,1)$-图和$(1,2)$-图算法作为预处理工具。由此得到$(2,2)$-图的$O(n^4(n+\min\{m,\overline m\})^3)$时间识别算法。