The exposition is the background on Johnny Dorset and the kidnappers, the rising action includes, in my view, the kidnapping of the boy and the two men locating to the remote cave.
The actual climax of the story occurs when the kidnappers are forced into submission by Johnny and a reversal of power takes place.
"During Sam's absence, the captor and the captive change roles, seemingly only in play but actually in real control of the situation. When Sam returns to the cave, he finds Bill, badly battered, playing the captured trapper to Johnny's heroic Indian, who calls himself “Red Chief.” Appropriating Sam for his game, Johnny announces that Bill is to be scalped and Sam burned at the stake."
During the confinement, Johnny abuses his kidnappers thoroughly, having a good time with his new playmates. The falling action occurs when the kidnappers make contact with Red Chief's father, looking for a ransom.
The resolution of the story is a reversal or a twist, very common in O Henry's work. The kidnappers, Bill and Sam must pay Johnny's father a sum of money so that he will take the boy off their hands.
"Johnny, however, does not wish to leave his captors. They must scheme to get him back to his father as once they had schemed to get him away from home, and finally, they must run at top speed to escape the boy who does not wish to lose his new playmates, the would-be kidnappers who have become his victims."