Sort and search: Exact algorithms for generalized domination
Résumé
In 1994, Telle introduced the following notion of domination, which generalizes many domination-type graph invariants. Let $\sigma$ and $\varrho$ be two sets of non negative integers. A vertex subset $S\subseteq V$ of an undirected graph $G=(V,E)$ is called a $(\sigma,\varrho)$-dominating set of $G$ if $|N(v)\cap S| \in \sigma$ for all $v\in S$ and $|N(v)\cap S| \in \varrho$ for all $v\in V\setminus S$. In this paper, we prove that decision, optimization, and counting variants of $(\{p\},\{q\})$-domination are solvable in time $2^{|V|/2}\cdot |V|^{O(1)}$. We also show how to extend these results for infinite $\sigma=\{p+m\cdot \ell\colon \ell\in \mathbb{N}_0\}$ and $\varrho=\{q+m\cdot \ell \colon \ell\in \mathbb{N}_0\}$. For the case $|\sigma|+|\varrho|=3$, these problems can be solved in time $3^{|V|/2}\cdot|V|^{O(1)}$, and similarly to the case $|\sigma|=|\varrho|=1$ it is possible to extend the algorithm for some infinite sets.