Can a breakpoint graph be decomposed into none other than 2-cycles?

Breakpoint graph has been widely used as a key data structure in algorithm design for genome rearrangements. The problem of breakpoint graph cycle decomposition, which asks for a largest collection of edge-disjoint cycles, is crucial in computing rearrangement distances between genomes. This problem is NP-hard, and can be approximated to 1.4193+ϵ. It is still open for deciding whether a breakpoint graph can admit a cycle decomposition with none other than 2-cycles. In this paper, we present a linear time algorithm to detect whether a breakpoint graph can be decomposed into none other than 2-cycles.