Faster Algorithms for Graph Monopolarity

A graph $G = (V,E)$ is $\textit{monopolar}$ if its vertex set admits a partition $V = (C \uplus{} I)$ where $G[C]$ is a $\textit{cluster graph}$ and $I$ is an $\textit{independent set}$ in $G$; this is a \textit{monopolar partition} of $G$. The MONOPOLAR RECOGNITION problem -- deciding whether an input graph is monopolar -- is known to be NP-Hard in very restricted graph classes such as sub-cubic planar graphs. We derive a polynomial-time algorithm that takes (i) a graph $G=(V,E)$ and (ii) a vertex modulator $S$ of $G$ to chair-free graphs as inputs, and checks whether $G$ has a monopolar partition $V=(C\uplus{}I)$ where set $S$ is contained in the cluster part. We build on this algorithm to develop fast exact exponential-time and parameterized algorithms for MONOPOLAR RECOGNITION. Our exact algorithm solves MONOPOLAR RECOGNITION in $\mathcal{O}^{\star}(1.3734^{n})$ time on input graphs with $n$ vertices, where the $\mathcal{O}^{\star}()$ notation hides polynomial factors. In fact, we solve the more general problems MONOPOLAR EXTENSTION and LIST-MONOPOLAR PARTITION in $\mathcal{O}^{\star}(1.3734^{n})$ time. These are the first improvements over the trivial $\mathcal{O}^{\star}(2^{n})$-time algorithms for all these problems. It is known that -- assuming ETH -- these problems cannot be solved in $\mathcal{O}^{\star}(2^{o(n)})$ time. Our FPT algorithms solve MONOPOLAR RECOGNITION in $\mathcal{O}^{\star}(3.076^{k_{v}})$ and $\mathcal{O}^{\star}(2.253^{k_{e}})$ time where $k_{v}$ and $k_{e}$ are, respectively, the sizes of the smallest vertex and edge modulators of the input graph to claw-free graphs. These results are a significant addition to the small number of FPT algorithms currently known for MONOPOLAR RECOGNITION.