Havens, a towering man of 70, has spent much of his career breeding cattle and owns a chain of Play Haven day-care centers. He is best known as the originator of the puggle, a pug-beagle cross with an irresistibly wrinkled muzzle, forlorn eyes and suitable dimensions for cramped city apartments. He first marketed puggles 20 years ago, but by late 2005, the dog suddenly had a cadre of celebrity owners, four-figure price tags and a brimming portfolio of magazine write-ups and morning-TV appearances. Puggle-emblazoned messenger bags and ladies’ track suits followed. For a time, in New York especially, you couldn’t swing a cat without hitting a puggle.

So-called designer dogs became popular a decade ago, beginning with the Labradoodle and other poodle crosses that sought to affix the poodle’s relatively nonshedding coat to other breeds. But the puggle, a designer dog with no clear design objective, seems to have set off an almost unintelligible free-for-all. Pugs alone are now being bred to Yorkshire terriers, Shih Tzus, bichon frisés, Pekingese, rat terriers, Boston terriers, dachshunds, Jack Russell terriers and Chihuahuas to create, respectively, Pugshires, Pug-Zus, Pushons, Puginese, Puggats, Pugstons, Daugs, Jugs and Chugs. Beagles mount Bostons. Chihuhuauas do Yorkies. Beagles and basset hounds are making Bagels; bassets and Shar-Peis are making Sharp Assets — “a more laid-back dog that says, ‘If you don’t feel like taking me for a walk, no big deal,’ ” Havens’s Web site claims. Poodles are being pushed further into a goofy taxonomy of portmanteau labels: Maltipoos, Eskipoos, Doodleman Pinschers. continued...