A Study of the Decycling Problem for the Generalized Exchanged Hypercube

The decycling number of a graphG is the size of a smallest set of vertices in G, whose removal leads to a survival graph with no cycles, that is, a forest. It is one of the earlier problems that was shown to be NP-Hard in 1972. From its definition, one can see that it has applications in reliability analysis of interconnection networks. The class of hypercubes is one of the most studied interconnection networks and there are bounds of the decycling numbers of hypercubes. Although the hypercube is a good example of interconnection networks, it has known drawbacks. The exchanged hypercube was introduced in 2005 to address some of these issues. Later, the generalized exchanged hypercube was introduced to include the exchange hypercube as a special case. In this paper, we obtain several upper bounds of the decycling number of the generalized exchanged hypercube, utilizing results for the hypercube.