Efficient recognition algorithms for $(1,2)$-, $(2,1)$- and $(2,2)$-graphs

A graph $G$ is said to be a $(k,\ell)$-graph if its vertex set can be partitioned into $k$ independent sets and $\ell$ cliques. It is well established that the recognition problem for $(k,\ell)$-graphs is NP-complete whenever $k \geq 3$ or $\ell \geq 3$, while it is solvable in polynomial time otherwise. In particular, for the case $k+\ell \leq 2$, recognition can be carried out in linear time, since split graphs coincide with the class of $(1,1)$-graphs, bipartite graphs correspond precisely to $(2,0)$-graphs, and $(\ell,k)$-graphs are the complements of $(k,\ell)$-graphs. Recognition algorithms for $(2,1)$- and $(1,2)$-graphs were provided by Brandst\"adt, Le and Szymczak in 1998, while the case of $(2,2)$-graphs was addressed by Feder, Hell, Klein, and Motwani in 1999. In this work, we refine these results by presenting improved recognition algorithms with lower time complexity. Specifically, we reduce the running time from $O((n+m)^2)$ to $O(n^2+nm)$ for $(2,1)$-graphs, from $O((n+\overline{m})^2)$ to $O(n^2+n\overline{m})$ for $(1,2)$-graphs, and from $O(n^{10}(n+m))$ to $O(n^4 (n+\min\{m,\overline{m}\})^3)$ for $(2,2)$-graphs. Here, $n$ and $m$ denote the number of vertices and edges of the input graph $G$, respectively, and $\overline{m}$ denotes the number of edges in the complement of $G$.