The vertex cover problem is the basis for de novo genome assembly using the overlap-consensus method. Assemblers like CABOG, the Celera assembler, construct an overlap graph of many whole-genome shotgun reads, where each vertex correspond to a read and the edges represent mutual overlap between two reads. The assembler then attempts to find a Hamiltonian path through the graph, where each vertex (read) is visited exactly once. The Hamiltonian path then represents the consensus sequence of the generated assembly.
Despite the effectiveness of the overlap-consensus method for long reads (+800 bp), De Bruijn assemblers have recently become the norm for short-read assembly. These assemblers trace an Eulerian path through a De Bruijn graph, where each edge corresponds to a kmer derived from each read and a vertex represents an exact overlap between two kmers.
Read more about genome assembly here. (There is a lot of published literature on this topic, most of it accessible through the links.)