2-domination in vague graphs

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) 2-dominating set and (perfect-total) 2-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 2-dominating set is a minimal perfect 2-dominating set and minimal dominating set. then, the concepts of (total) 2-cobondage set and (total) 2-cobondage number in vague graphs are expressed and related results obtained. finally, an application related to fire stations and emergency medical centers is provided.