If either of two adjoining landowners does not need a fence, the landowner that needs a fence may build the entire fence and report the total cost to the associate circuit judge who shall authorize the cost to be recorded on each deed. Should the landowner that claimed no need for a fence subsequently place livestock against the fence, the landowner that built the fence shall be reimbursed for one-half the construction costs share to be determined as provided in section 272.060.