If a party owns the rights to a particular trademark, that party can sue subsequent parties for trademark infringement. 15 U.S.C. 1114, 1125. Bear in mind that a trademark is generally a symbol, picture, or other signature identification of the business.
The standard for trademark infringement is whether there is a "likelihood of confusion." To be more specific, the use of a trademark in connection with the sale of a good constitutes infringement if it is likely to cause consumer confusion as to the source of those goods or as to the sponsorship or approval of such goods.
In deciding whether consumers are likely to be confused, the courts will typically look to a number of factors, including: (1) the strength of the mark; (2) the proximity of the goods; (3) the similarity of the marks; (4) evidence of actual confusion; (5) the similarity of marketing channels used; (6) the degree of caution exercised by the typical purchaser; (7) the defendant's intent. Polaroid Corp. v. Polarad Elect. Corp., 287 F.2d 492 (2d Cir.), cert. denied, 368 U.S. 820 (1961).
The use of an identical mark on the same product would clearly constitute infringement. If I manufacture and sell computers using the mark "Apple," my use of that mark will likely cause confusion among consumers, since they may be misled into thinking that the computers are made by Apple Computer, Inc. Using a very similar mark on the same product may also give rise to a claim of infringement, if the marks are close enough in sound, appearance, or meaning so as to cause confusion. So, for example, "Applet" computers may be off-limits; perhaps also "Apricot." On the other end of the spectrum, using the same term on a completely unrelated product will not likely give rise to an infringement claim. Thus, Apple Computer and Apple Records can peacefully co-exist, since consumers are not likely to think that the computers are being made by the record company, or vice versa.
Between the two ends of the spectrum lie many close cases, in which the courts will apply the factors listed above. So, for example, where the marks are similar and the products are also similar, it will be difficult to determine whether consumer confusion is likely. In one case, the owners of the mark "Slickcraft" used the mark in connection with the sale of boats used for general family recreation. They brought an infringement action against a company that used the mark "Sleekcraft" in connection with the sale of high-speed performance boats. Because the two types of boats served substantially different markets, the court concluded that the products were related but not identical. However, after examining many of the factors listed above, the court concluded that the use of Sleekcraft was likely to cause confusion among consumers. AMF Inc. v. Sleekcraft Boats, 599 F.2d 341 (9th Cir. 1979).
One thing is that if Pet Depot is not in business in your state, you'll have a better position. However, this is a slippery issue and I'd advise that you speak to an attorney in your area about proceeding with this before you proceed.
Source/more reading: http://cyber.law.harvard.edu/metaschool/fisher/domain/tm.htm#7
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