Sadegh Banitalebi - Department of Knowledge and Cognitive Intelligence, Imam Hossein University, Tehran, Iran.
Rajab Ali Borzooei - Department of Mathematics, Shahid Beheshti University, Tehran, Iran.
Elahe Mohamadzadeh - Department of Mathematics, Faculty of Science, Payam Noor University, Tehran, Iran.

A vague graph is a generalized structure of a fuzzy graph that gives more precision, flexibility and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, the notions of (perfect-total) ۲-dominating set and (perfect-total) ۲-domination numbers on vague graphs are introduced and some properties are investigated. Especially, it is proven that in any strong vague graph on a Petersen graph, any minimal ۲-dominating set is a minimal perfect ۲-dominating set and minimal dominating set. Then, the concepts of (total) ۲-cobondage set and (total) ۲-cobondage number in vague graphs are expressed and related results obtained. Finally, an application related to Fire Stations and Emergency Medical centers is provided.

Vague graph, ۲-dominating set, ۲-cobondage set