Convexity and starshapedness of feasible sets in stationary flow networks

Figure 1.  Example of differently structured graphs

Figure 2.  Example for illustrating the notation (graph numbered by breadth-first search)

Figure 3.  Example for illustrating the triple and quadruple indices on the path from $ G_1 $ to $ G_5 $

Figure 4.  Linear graph of example 1

Figure 5.  Feasible sets for different pressure bounds

Figure 6.  Linear graph of example 2

Figure 7.  Set $ M_{a} $ for $ (p^{+, \min})_{a} = [1, 1, 1, 1]^T $ and $ (p^{+, \max})_{a} = [3, 3, 3, 3]^T $

Figure 8.  Set $ M_{b} $ for $ (p^{+, \min})_{b} = [2.5, 2, 1.5, 1]^T $ and $ (p^{+, \max})_{b} = [3, 2.5, 2, 1.5]^T $

Figure 9.  Linear graph with one compressor edge of example 3

Figure 10.  Set $ M_{a} $ for $ (p^{+, \min})_{a} = [1, 1, 1, 1]^T $ and $ (p^{+, \max})_{a} = [3, 3, 3, 3]^T $

Figure 11.  Set $ M_{b} $ for $ (p^{+, \min})_{b} = [2.5, 2, 1.5, 1]^T $ and $ (p^{+, \max})_{b} = [3, 2.5, 2, 1.5]^T $

Figure 12.  Graph of example 2

Figure 13.  Feasible sets for different pressure bounds

Figure 14.  Graph of example 5

Figure 15.  Set $ M_{a} $ for $ (p^{+, \min})_{a} = [1, 1, 1, 1]^T $ and $ (p^{+, \max})_{a} = [3, 2, 2, 2]^T $

Figure 16.  Set $ M_{a} $ for $ (p^{+, \min})_{b} = [2, 1, 1, 1]^T $ and $ (p^{+, \max})_{b} = [3, 2, 2, 1.5]^T $