Open-separating dominating codes in graphs
<|reference_start|>Open-separating dominating codes in graphs: Using dominating sets to separate vertices of graphs is a well-studied problem in the larger domain of identification problems. In such problems, the objective is to choose a suitable dominating set $C$ of a graph $G$ such that the neighbourhoods of all vertices of $G$ have distinct intersections with $C$. Such a dominating and separating set $C$ is often referred to as a \emph{code} in the literature. Depending on the types of dominating and separating sets used, various problems arise under various names in the literature. In this paper, we introduce a new problem in the same realm of identification problems whereby the code, called \emph{open-separating dominating code}, or \emph{OSD-code} for short, is a dominating set and uses open neighbourhoods for separating vertices. The paper studies the fundamental properties concerning the existence, hardness and minimality of OSD-codes. Due to the emergence of a close and yet difficult to establish relation of the OSD-codes with another well-studied code in the literature called open locating dominating codes, or OLD-codes for short, we compare the two on various graph families. Finally, we also provide an equivalent reformulation of the problem of finding OSD-codes of a graph as a covering problem in a suitable hypergraph and discuss the polyhedra associated with OSD-codes, again in relation to OLD-codes of some graph families already studied in this context.<|reference_end|>
@article{chakraborty2024open-separating,
    title={Open-separating dominating codes in graphs},
    author={Dipayan Chakraborty and Annegret K. Wagler},
    journal={arXiv preprint arXiv:2402.03015},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.03015},
    primaryClass={math.CO cs.DM}
}