Mogollon Indian pictograph of Tularosa Valley, New Mexico. The Mogollon had a more than commonly sophisticated and ornate style for their art. This checkerboard dog is not duplicated by any other people. It may have a specific though lost symbolic meaning, or the dog is perhaps merely wearing a woven winter jacket!
It might also just be a tic-tac-toe type game associated with some myth involving sun, moon, and dog, with the game pieces (sun and moon) also shown in the pictograph -- for all Native American games had associated myths.