Two Applications of Graph Minor Reduction

In this paper, we study two applications of graph minor reduction. In the first part of the paper, we introduce a variant of the boxicity, called strong boxicity, where the rectangular representation satisfies an additional condition that each rectangle contains at least one point not present in any other rectangle. We show how the strong boxicity of a graph~$G$ can be estimated in terms of the strong boxicity of a minor~$H$ and the number of edit operations needed to obtain~$H$ from~$G.$ In the second part of the paper, we consider false data injection (attack) in a flow graph~$G$ and quantify the subsequent effect on the state of edges of~$G$ via the \emph{edge variation factor}~$\theta.$ We use minor reduction techniques to obtain bounds on~$\theta$ in terms of the connectivity parameters of~$G,$ when the attacker has complete knowledge of~$G$ and also discuss stealthy attacks with partial knowledge of the flow graph.