The robber's first story is about his attempt to steal "an immense treasure of many thousand pieces of gold and silver" from a giant. With his men, he goes to plunder the treasure, but the giant returns and catches them. The robber and his hundred men are divided among ten of the giants so that there are groups of ten robbers for each giant. The giant from whom they were trying to steal takes the robber and nine other men to a cave where he eats one a day. The robber - as the thinnest of the ten - is left for last. The robber notices that the giant has bad eyes, so he starts scheming. He claims he is a physician and can help the giant. Then he "put[s] oil in a vessel and mixed in with sulphur, pitch, salt, arsenic, and other destructive ingredients, and then I put it over the fire, as if I were preparing a plaster for his eyes." The giant loses his sight from the robber's treatment and storms around his cave, striking the floor with a club. The robber, however, is still trapped, as "the cave was everywhere surrounded with high walls, and the doors were closed with iron bolts." The robber disguises himself as a ram among the giant's sheep, which works for awhile, but then the giant picks the fattest of his flock (the robber in disguise) to eat. The robber jumps out of the giant's hand and eventually escapes, "shout[ing] in a mocking tone to him that I had escaped him in spite of all."
There are more than a few similarities between the robber's story and Odysseus' experience with the Cyclops in Book IX of The Odyssey. [The following quotes are from E.V. Rieu's translation.]
While Odysseus and his men don't have the intention to steal from the Cyclops (Odysseus even prevents his men from taking the Cyclop's cheeses: "I was not to be persuaded. I wished to see the owner of the cave and had hopes of some friendly gifts from my host."), they do take "some cheeses just for ourselves" and eat them while waiting for the Cyclops to return from tending his flock.
When the Cyclops returns, he seals the cave with a stone. "It was a mighty slab, such as you couldn't have budged from the ground, not with a score of heavy four-wheeled waggons to help you." The Cyclops doesn't show hospitality to Odysseus and his men, rather he
seized a couple [of the men] and dashed their heads against the floor as though they had been puppies. Their brains ran out of the ground and soaked the earth. Limb by limb he tore them to pieces to make his meal, which he devoured like a mountain lion, never pausing till entrails and flesh, marrow and bones, where all consumed.The next day, the Cyclops "once more seized upon two of us and prepared his supper." But meanwhile, Odysseus has been scheming. With his men, he's sharpened a piece of timber, "poked it into the blazing fire to make it hard," and plans to "lift the pole and twist it in the Cyclops' eye when he was sound asleep." Odysseus encourages the Cyclops to drink some wine and tells him that his name is Nobody, to which the Cyclops replies, "Of all his company I will eat Nobody last, and the rest before him. That shall be your gift."
Odysseus' plan works, and they skewer the Cyclops' eye. "The fiery smoke from the blazing eyeball singed his lids and brow all round, and the very roots of his eye crackled in the heat." After complaining to his friends that "Nobody's treachery... is doing me to death," the Cyclops opens the cave and plans to catch Odysseus and company as they run out. Odysseus ties each of his men beneath three sheep, but he hides himself under a single ram. The Cyclops doesn't notice the men beneath the sheep, but he becomes suspicious about the ram, asking "Why are you the last of the flock to pass out of the cave, you who have never lagged behind the sheep?" Eventually, he lets the ram go, and Odysseus frees himself and his men from their sheepy bonds.
Odysseus has to taunt the Cyclops though, even after his crew warn him not to. While still in range of the Cyclops' hearing, Odysseus shouts back at him: "Cyclops, if anyone ever asks you how you came by your unsightly blindness, tell him your eye was put out by Odysseus, Sacker of Cities, the son of Laertes, who lives in Ithaca."
Both stories involve an attempt at robbery, a giant, an almost-inescapable cave, the daily eating of the narrator's men (with the narrator saved for last), the putting out of an eye, hiding under a ram amid a flock of sheep, and a final taunting. After each final taunting, the narrator is almost caught a second time. In "The Robber and His Sons," the robber is given a ring that causes him to repeatedly say, "Here I am!" and in The Odyssey, Odysseus' shout gives the Cyclops a target to throw a rock at.
I think it's pretty clear that the story in "The Robber and His Sons" took at least a few elements from The Odyssey.
