Can A Breakpoint Graph Be Decomposed Into None Other Than 2-Cycles?

Breakpoint graph is a key data structure to study genome rearrangements. The problem of Breakpoint Graph Decomposition (BGD), which asks for a largest collection of edge-disjoint cycles in a breakpoint graph, is a crucial step in computing rearrangement distances between genomes. This problem for genomes of unsigned genes is proved NP-hard, and the best known approximation ratio is 1.4193+is an element of [1]. In this paper, we present a polynomial time algorithm to detect whether a breakpoint graph can be decomposed into none other than 2-cycles. Our algorithm can be used to detect if there exists a sorting scenario between two genomes without reusing any breakpoints.