From Mitch4:
He notes,
Huh? I can’t even identify the additional object next to the playing cards — nail clipper? cigarillo? — let alone what the dog is supposed to do with them to become more adoptable.
I think it’s a cigar–pretty sure that’s a cigar band in the middle of the object–but what do cards and cigars have to do with getting adopted?!
At the risk of being called mean, that strip bills itself as “REMIXED NEW YORKER CARTOON REJECTS”. I’d submit that if your cartoon is too obscure for TNY, you might have a problem!
Comics I Don't Understand 
I think it’s a reference to the “Dogs Playing Poker” painting.
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That seems likely, Mark, but I can’t figure out what that has to do with getting adopted.
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Some people think painting of dogs playing poker are cute, and cute gets dogs adopted.
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Already answered, but . . .
I always thought this was just a one-off kitschy painting, but as it turns out, it’s a whole series, and many were created to advertise for cigars.
https://en.wikipedia.org/wiki/Dogs_Playing_Poker
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