Global relabeling heuristic: Push-relabel maxflow
Q1: The invariant goes the other way around: height[n1] &lt;= height[n2] + 1 for all edges (n1,n2) in the residual graph.

As the sink (10) is not adjacent from (7) in the residual graph, the configuration height[7] == 14, height[10] == 0 is perfectly fine.

The other way around, (7) is adjacent from (10) in the residual graph, and indeed height[10] &lt;= height[7] + 1.