February 17, 2000
The LP Matrix Partition Theorem and its use with Fractional Domination and Domatic Parameters
Many graph theoretic subset parameters, such as fractional domination, fractional independence, and fractional packing, can be formulated as Linear Programming (LP) problems, where the constraint matrix for the LP-problem is the closed neighborhood, adjacency, or incidence matrix. The Automorphism Class Theorem (ACT) states that there exists an optimum LP-solution which is constant on automorphism classes. We investigate the properties of the constraint matrix which allow for this constant solution. The property that results is described as the Partition Class Theorem (PCT). We show that the PCT is a generalization of the ACT. As an extension of the fractional domination and fractional domatic graphical parameters, multi-fractional domination is introduced. We will demonstrate its LP-formulation, and to this formulation we apply the PCT. We investigate some properties of the multi-fractional domination number and its relationship to the fractional domination and fractional domatic numbers. We introduce and investigate the (r, s)-fractional domatic number, a generalization of the domatic number, r-configuration number, and the fractional domatic number.