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.

翻译：利用支配集分离图中的顶点，是识别问题领域中一个被广泛研究的问题。此类问题的目标是为图$G$选择一个合适的支配集$C$，使得$G$中所有顶点的邻域与$C$的交集互不相同。这种兼具支配和分离性质的集合$C$在文献中常被称为"码"（code）。根据所采用的支配集和分离集类型的不同，相关文献以多种名称提出了各种不同的问题。本文在这一识别问题框架下引入了一个新问题：该码被称为"开分离支配码"（简称OSD-码），它是一个支配集，并利用开邻域来分离顶点。本文研究了OSD-码的存在性、难解性和极小性等基本性质。由于OSD-码与文献中另一类被广泛研究的码（称为"开定位支配码"，简称OLD-码）之间存在紧密但难以建立的联系，本文在多种图族上对两者进行了比较。最后，我们还给出了将寻找图的OSD-码问题等价转化为适当超图中的覆盖问题的方法，并讨论了与OSD-码相关联的多面体，再次将其与已经在相关上下文中研究过的某些图族的OLD-码进行了关联。