Maybe. It depends whether the fence is on the actual boundary lines between the properties or whether the fence is constructed entirely on your neighbor's land. It also depends on the way you and your neighbor intend to use your separate properties. Generally, a landowner has a right to build a fence on her own property at her own expense. Even if it is close to the property line, as long as it is not on the boundary this type of fence is not a true partition fence and you have no rights to it nor any duties to maintain your neighbor's fence. If you and your neighbor desire a true partition fence to divide the properties, you can voluntarily agree to share the costs if you think the fence would also be to your benefit. This would ensure that you have a say in how it is constructed and maintained, and that it is not removed without your consent.
West Virginia has statutes that govern partition fences in rural areas. Owners must share the cost of a partition fence if the land is used for “agricultural, horticultural, grazing, or livestock purposes.” This includes all construction, repair, and maintenance costs. If your neighbor builds a fence at her own cost because you were not using your land for the listed purposes at the time of the construction but you later decide to use your land in such a way, you are responsible to your neighbor for half of the cost of the partition fence. These laws can be overridden by a contractual agreement with your neighbor.
Additionally, West Virginia has a special statute for hedge fences. A hedge fence may not extend more than 18 inches over the dividing line of a neighbor's property or into a public road. A violation of this statute could result in a 10 day notice to correct the problem. If the hedge is not trimmed after the 10 days, it is considered a misdemeanor and the owner can be fined up to $1.00 every day until it is trimmed.
For more information, see: W. Va. Code § 19-17-3 to -8 (2011); Cutright v. Sexton, 99 W. Va. 69 (1924); Doyle, D., Smith, D., Ferrise, A., Real Property: Landowners’ Rights and Responsibilities in West Virginia,
http://www.wvu.edu/~exten/infores/pubs/other/rd702.pdf (last visited June 19, 2012).