Algorithms for Minimum Membership Dominating Set Problem
<|reference_start|>Algorithms for Minimum Membership Dominating Set Problem: Given a graph $G = (V, E)$ and an integer $k$, the Minimum Membership Dominating Set problem asks to compute a set $S \subseteq V$ such that for each $v \in V$, $1 \leq |N[v] \cap S| \leq k$. The problem is known to be NP-complete even on split graphs and planar bipartite graphs. In this paper, we approach the problem from the algorithmic standpoint and obtain several interesting results. We give an $\mathcal{O}^*(1.747^n)$ time algorithm for the problem on split graphs. Following a reduction from a special case of 1-in-3 SAT problem, we show that there is no sub-exponential time algorithm running in time $\mathcal{O}^*(2^{o(n)})$ for bipartite graphs, for any $k \geq 2$. We also prove that the problem is NP-complete when $\Delta = k+2$, for any $k\geq 5$, even for bipartite graphs. We investigate the parameterized complexity of the problem for the parameter twin cover and the combined parameter distance to cluster, membership($k$) and prove that the problem is fixed-parameter tractable. Using a dynamic programming based approach, we obtain a linear-time algorithm for trees.<|reference_end|>
@article{reddy2024algorithms,
    title={Algorithms for Minimum Membership Dominating Set Problem},
    author={Sangam Balchandar Reddy and Anjeneya Swami Kare},
    journal={arXiv preprint arXiv:2408.00797},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.00797},
    primaryClass={cs.DS cs.CC}
}