Graph relabeling with stacked labels

This paper describes a new problem in graph theory called the GRAPH RELABELING WITH STACKED LABELS. Given a simple and connected graph G = (V, E), two labelings L and Ľ of G, the problem is to make a series of transformation from ≪G, L≫ to ≪G, Ľ ≫, where ≪G, L≫ is the graph G with labeling L. The transformation in consideration here is a flip operation. A flip operation allows a pair of stacked labels in two adjacent vertices to exchange places between vertices in a certain fashion. In this paper we show that this problem in general is insolvable. We precisely characterize the solvability for this problem when G is either a path graph or a tree and in the process we also have polynomial time algorithms to solve the problem in both cases. Additionally, we also show that our algorithm is exact and provably fastest in the case G is a path graph. Potential applications and open problems are also discussed.