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 $G$ which is also separating in the sense that the neighbourhoods of any two distinct vertices of $G$ have distinct intersections with $C$. Such a dominating and separating set $C$ of a graph is often referred to as a 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 open-separating dominating code, or OD-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 OD-codes. Due to the emergence of a close and yet difficult to establish relation of the OD-code with another well-studied code in the literature called open (neighborhood)-locating dominating code (referred to as the open-separating total-dominating code and abbreviated as OTD-code in this paper), we compare the two codes on various graph families. Finally, we also provide an equivalent reformulation of the problem of finding OD-codes of a graph as a covering problem in a suitable hypergraph and discuss the polyhedra associated with OD-codes, again in relation to OTD-codes of some graph families already studied in this context.

翻译：利用支配集分离图顶点是识别问题这一更大领域中一个被深入研究的问题。在此类问题中，目标是为图G选择一个合适的支配集C，该集合同时具有分离性，即图G中任意两个不同顶点的邻域与C的交集互不相同。这种兼具支配性和分离性的图集合在文献中通常被称为码。根据所使用的支配集和分离集类型的不同，文献中出现了各种以不同名称命名的问题。本文在识别问题的同一领域内引入了一个新问题，其中被称为开放分离支配码（简称OD码）的码是一个支配集，并利用开放邻域来分离顶点。本文研究了关于OD码的存在性、困难性和极小性的基本性质。由于OD码与文献中另一种被深入研究的码——开放（邻域）定位支配码（本文中称为开放分离全支配码，简称OTD码）之间出现了一种密切但难以建立的关系，我们在多种图族上比较了这两种码。最后，我们还将寻找图OD码的问题等价地重新表述为在合适超图中的覆盖问题，并讨论了与OD码相关的多面体，同样是与一些已在此背景下研究过的图族的OTD码相关联。