Figure 2. A. Graph studied by Schapiro et al. [41], showing the optimal decomposition.

B. Task display from Experiment 1. Participants used the computer mouse to select three locations adjacent to the probe location. C. Graph employed in Experiment 1, showing the optimal decomposition. Width of each gray ring indicates mean proportion of cases in which the relevant location was chosen. D. Graph studied in Experiments 2 and 3, showing the optimal decomposition (two regions, with central vertex grouped either to left or right). Top: Illustration of a “delivery” assignment from Experiment 3 (green = start, red = goal), where bottleneck (purple) and non-bottleneck (blue) probes called for a positive response. Bottom: An assignment where bottleneck and non-bottleneck probes called for a negative response. E. Mean correct response times from Experiment 3. Affirm: trials where the probe fell on the shortest path between the specified start and goal locations. Reject: trials where it did not. Purple: bottleneck probes. Blue: non-bottleneck probes. F. State-transition graph for the Tower of Hanoi puzzle, showing the optimal decomposition and indicating the start and goal configurations of the kind studied in Experiment 4. A different set of colors was used for the beads in the actual experiment. Furthermore, as explained under Methods, the beads were the same size. The changes were made here for display purposes.