(p,q)-Graceful signed graph

[math]\displaystyle{ (p,q) }[/math]-Graceful signed graph --- [math]\displaystyle{ (p,q) }[/math]-грациозный знаковый граф.

Let [math]\displaystyle{ S = (G,s) }[/math] be a sigraph and [math]\displaystyle{ s }[/math] be a function which assigns a positive or a negative sign to each edge of [math]\displaystyle{ G }[/math]. Let the sets [math]\displaystyle{ E^{+} }[/math] and [math]\displaystyle{ E^{-} }[/math] consist of [math]\displaystyle{ m }[/math] positive and [math]\displaystyle{ n }[/math] negative edges of [math]\displaystyle{ G }[/math], respectively, where [math]\displaystyle{ m+n = q }[/math]. Given positive integers [math]\displaystyle{ k }[/math] and [math]\displaystyle{ d }[/math], [math]\displaystyle{ S }[/math] said to be [math]\displaystyle{ (k,d) }[/math]-graceful if the vertices of [math]\displaystyle{ G }[/math] can be labeled with distinct integers from the set [math]\displaystyle{ \{0, 1, \ldots, k+(q-1)d\} }[/math] such that, when each edge [math]\displaystyle{ uv }[/math] of [math]\displaystyle{ G }[/math] is assigned the product of its sign and the absolute difference of the integers assigned to [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math], the edges in [math]\displaystyle{ E^{+} }[/math] and [math]\displaystyle{ E^{-} }[/math] are labeled with [math]\displaystyle{ k, k+d, k+2d, \ldots, k+(m-1)d }[/math] and [math]\displaystyle{ -k, -(k+d), -(k+2d), \ldots, -(k+(n-1)d) }[/math], respectively.