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arXiv · 2402.03015

On open-separating dominating codes in graphs

Dipayan Chakraborty·Annegret K. Wagler
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Abstract

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.

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BibTeXRIS

Dipayan Chakraborty, Annegret K. Wagler. 2024-02-05. On open-separating dominating codes in graphs. https://doi.org/10.1016/j.dam.2025.05.045

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