Network encoding complexity: Exact values, bounds, and inequalities

Figure 1.  Paths $\beta_1, \beta_2$ merge at edge $A \to B$ and at merged subpath (or merging) $A \to B \to C \to D$, and paths $\beta_1, \beta_2, \beta_3$ merge at edge $B \to C$ and at merged subpath (or merging) $B\to C$ (Here, arrows in the figure represents edges, and the terminal points of arrows should be naturally interpreted as vertices; the same convention applies to other figures in this paper)

Figure 2.  (a) Network coding in the butterfly network (b) Network coding in a two-way channel (c) Network coding in two sessions of unicast

Figure 3.  An example of a reroutable graph

Figure 4.  Two examples of merging sequences (as in Remark 3.3, all the mergings in this figure are represented by solid dots instead)

Figure 5.  Two examples of alternating sequences

Figure 6.  (a) A non-reroutable (2, 3)-graph with 8 mergings (b) An example of a (2, 5)-graph

Figure 7.  Graph $\mathcal{E}(4, 4)$ with $9$ mergings

Figure 8.  (a) Graph $\mathcal{F}(3, 3)$ with $11$ mergings (b) Splitting of $R_1$ in $\mathcal{F}(3, 3)$

Figure 9.  Concatenation of $\mathcal{F}(2, 2)$ and a non-reroutable $(2, 2)$-graph

Figure 10.  Partition a $(3, c_2)$-graph into blocks

Figure 11.  Case 1

Figure 12.  Case 2

Figure 13.  Concatenation of two (3, 3)-graphs

Figure 14.  Concatenation of a (2, 2)-graph and a (4, 4)-graph