The traditional dynamic resource location problem attempts to minimize the cost of servicing a number of sequential requests, given foreknowledge of a limited number of requests. One artificial constraint of this problem is the presumption that resource relocation and remote servicing of requests have identical costs. Parameterizing the ratio of relocation cost to service cost leads to two extreme behaviors in terms of dynamic optimizability. The threshold at which a specific graph transitions between these behaviors reveals certain characteristics of the graph's decomposability into cycles.