Figure 3
From: A virtual pebble game to ensemble average graph rigidity
Illustration of how to play the VPG on a network with 3 vertices. A dashed line denotes the edge that is being added at the current step, whose capacity is indicated in a dashed box. Appearing at the ends of an edge is the number of pebbles consumed from the corresponding vertex used to cover the edge. The operations that rearrange pebbles are described as a sequence of labeled steps. Numbers listed in dashed boxes refer to pebble capacities of an edge that is to be placed, but not yet placed in the network until the required number of free pebbles can be collected. (a-b) Each vertex is initially assigned 6 pebbles, and an edge of capacity 2.5 is added to the graph. (c) Vertex v 2 has 3.5 free pebbles and cannot fully cover the new edge between v 2 and v 3 (which requires 5 pebbles). A pebble search is carried out and 1.5 pebbles are backtracked through the edge between v 1 and v 2. (d-e) Vertex v 1 has enough free pebbles to cover the newly added edge. (f) Adding an additional edge between v 2 and v 3, the two edges (of capacity 5 and 1.5) combine, yielding a partially covered edge with capacity 6.5. Of course physically, the greatest possible covering is 6. As such, only six pebbles can cover the edge. (g) Because the edge between v 2 and v 3 cannot be fully covered, the attempted pebble search fails, which leads to the condensation of v 2 and v 3 into a single vertex denoted as v 2. (h) Edges v 1 − v 2 and v 1 − v 3 in step (g) are combined into one edge v 1 − v 2.