The Kill-Off is a murder mystery such as Agatha Christie wrote. We meet a character who is so unlikable we wish someone would murder her. We meet a small community of characters each of whom has a good reason to do the deed. When someone does, we try to figure out who did it, or, as fans say, whodunit. Obviously this is in the tradition of Murder at the Vicarage, Murder on the Orient Express and other Christie classics.
At first, it's hard to see the similarity because Thompson's setting has little in common with Christie's English country houses. There is no elegance, no sophistication. His fictional resort town is down on it's luck and populated by people just trying to get by. He wrote about the world he knew.
Though working in a familiar form, Thompson did something extraordinary in The Kill-Off: there is no sleuth. No equivalent of Miss Marple or Hercule Poirot, or, for that matter, Sherlock Holmes or Nero Wolfe. No genius to solve the mystery. Christie did this as well, though I am aware of only one example, And Then There Were None.
And Thompson did something I've never seen anywhere else. Each chapter is narrated in the first person by a different character---twelve chapters, twelve characters telling their own stories of past connections to the victim and others in town and telling where they were on the night of the murder . . . except when they're lying.