A method, comprising: providing a first plurality of double stranded target polynucleotide fragments, each fragment having a first end and a second end; providing a first adapter oligonucleotide comprising a first strand having a 5' end and a 3' end and a second strand having a 5' end and a 3' end, wherein the first adapter oligonucleotide comprises (i) a double stranded region comprising the 5' end of the first strand and the 3' end of the second strand, and (ii) a single stranded region in which the first and second strands are single stranded, wherein the single stranded region comprises the 3' end of the first strand and the 5' end of the second strand, wherein the first adapter oligonucleotide comprises a first library-specific sequence, wherein the 3' end of the first strand is modified to prevent digestion by an enzyme having 3' exonuclease activity and wherein the 5' end of the second strand is modified to prevent digestion by an enzyme having 5' exonuclease activity; incubating the first adapter oligonucleotide and the first plurality of double stranded target polynucleotide fragments under conditions suitable to ligate the 5' end of the first strand of the first adapter and the 3' end of the second strand of the first adapter to the first and second ends of the double stranded target polynucleotide fragments to produce a first library of polynucleotides comprising first adapter-target-first adapter sequences; and contacting the first library of polynucleotides with an exonuclease, wherein the exonuclease comprises 3' and 5' single stranded exonuclease activity, to selectively degrade first adapter oligonucleotides that are not ligated to the double stranded target polynucleotide fragments. providing a first plurality of double stranded target polynucleotide fragments, each fragment having a first end and a second end; providing a first adapter oligonucleotide comprising a first strand having a 5' end and a 3' end and a second strand having a 5' end and a 3' end, wherein the first adapter oligonucleotide comprises (i) a double stranded region comprising the 5' end of the first strand and the 3' end of the second strand, and (ii) a single stranded region in which the first and second strands are single stranded, wherein the single stranded region comprises the 3' end of the first strand and the 5' end of the second strand, wherein the first adapter oligonucleotide comprises a first library-specific sequence, wherein the 3' end of the first strand is modified to prevent digestion by an enzyme having 3' exonuclease activity and wherein the 5' end of the second strand is modified to prevent digestion by an enzyme having 5' exonuclease activity; wherein the first adapter oligonucleotide comprises (i) a double stranded region comprising the 5' end of the first strand and the 3' end of the second strand, and (ii) a single stranded region in which the first and second strands are single stranded, wherein the single stranded region comprises the 3' end of the first strand and the 5' end of the second strand, wherein the first adapter oligonucleotide comprises a first library-specific sequence, wherein the 3' end of the first strand is modified to prevent digestion by an enzyme having 3' exonuclease activity and wherein the 5' end of the second strand is modified to prevent digestion by an enzyme having 5' exonuclease activity; incubating the first adapter oligonucleotide and the first plurality of double stranded target polynucleotide fragments under conditions suitable to ligate the 5' end of the first strand of the first adapter and the 3' end of the second strand of the first adapter to the first and second ends of the double stranded target polynucleotide fragments to produce a first library of polynucleotides comprising first adapter-target-first adapter sequences; and contacting the first library of polynucleotides with an exonuclease, wherein the exonuclease comprises 3' and 5' single stranded exonuclease activity, to selectively degrade first adapter oligonucleotides that are not ligated to the double stranded target polynucleotide fragments.