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 typically to separate any two vertices of a graph by their unique neighbourhoods in a suitably chosen dominating set of the graph. Such a dominating and separating set 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 the \emph{open-separating dominating code}, or the \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 the open locating dominating codes, or OLD-codes for short, we compare the two on various graph classes. 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 classes already studied in this context.

翻译：利用控制集分离图顶点是识别问题领域中一个被广泛研究的问题。在此类问题中，目标通常是通过图中某个适当选取的控制集中顶点的唯一邻域来分离任意两个顶点。文献中通常将这样的控制与分离集称为“码”。根据所使用的控制集与分离集类型不同，文献中以不同名称衍生出多种问题。本文在该识别问题框架下引入一个新问题：该码称为“开放分离控制码”（简称OSD码），它是一个控制集，并利用开放邻域实现顶点分离。本文研究了OSD码在存在性、难解性和极小性方面的基本性质。由于OSD码与文献中另一个被广泛研究的码（称为开放定位控制码，简称OLD码）之间存在密切但难以建立的关系，我们在多种图类上对两者进行了比较。最后，本文将寻找图的OSD码问题等价地转化为适当超图中的覆盖问题，并讨论了与OSD码相关的多面体——再次联系到在该背景下已被研究的某些图类的OLD码。