ABSTRACT
For an arbitrary subset https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315141428/0b8b6f32-6ee7-486d-b7f9-109dc1ca019f/content/p.tif"/> of the reals, we define a function https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315141428/0b8b6f32-6ee7-486d-b7f9-109dc1ca019f/content/eqn41_1.tif"/> to be a https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315141428/0b8b6f32-6ee7-486d-b7f9-109dc1ca019f/content/p.tif"/>-dominating function of a graph G = (V, E) if the sum of its function values over any closed neighbourhood is at least 1. That is, for every v ∈ V, f(N(v) ∪ {v}) ≥ 1. The https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315141428/0b8b6f32-6ee7-486d-b7f9-109dc1ca019f/content/p.tif"/>-domination number of a graph G is defined to be the infimum of f(V) taken over all https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315141428/0b8b6f32-6ee7-486d-b7f9-109dc1ca019f/content/p.tif"/>-dominating functions f. When https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315141428/0b8b6f32-6ee7-486d-b7f9-109dc1ca019f/content/eqn31_1.tif"/> we obtain the standard domination number. When https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315141428/0b8b6f32-6ee7-486d-b7f9-109dc1ca019f/content/eqn31_2.tif"/>, {−1,0, 1} or {−1,1} we obtain the fractional, minus or signed domination numbers, respectively. In this chapter, we survey some recent results concerning dominating functions in which negative weights are allowed.
