bounds of unique response strong roman domination

Given a simple graph g = (v; e) with maximum degree ∆. let (v0; v1; v2) be an ordered partitionof v , where vi = fv 2 v : f(v) = ig for i = 0; 1 and v2 = fv 2 v : f(v) ≥ 2g. a function f :v ! f0; 1; :::; d∆2 e + 1g is a strong roman dominating function (strdf) on g, if every v 2 v0 has aneighbor w 2 v2 and f(w) ≥ 1 + d12jn(w) v0je. a function f : v ! f0; 1; :::; d∆2 e + 1g is a uniqueresponse strong roman function (urstrf), if w 2 v0, then jn(w) v2j ≤ 1 and w 2 v1 [v2 implies thatjn(w) v2j = 0. a function f : v ! f0; 1; :::; d∆2 e + 1g is a unique response strong roman dominatingfunction (urstrdf) if it is both urstrf and strdf. the unique response strong roman dominationnumber of g, denoted by ustr(g), is the minimum weight of a unique response strong roman dominatingfunction. in this paper we initiate the study of several mathematical properties of this invariant. weobtain several bounds on such a parameter and give some realizability results for it.