  Linear network coding transmits data through networks by letting the
intermediate nodes combine the messages they receive and forward the
combinations towards their destinations. The solvability problem asks whether
the demands of all the destinations can be simultaneously satisfied by using
linear network coding. The guessing number approach converts this problem to
determining the number of fixed points of coding functions $f:A^n\to A^n$ over
a finite alphabet $A$ (usually referred to as Boolean networks if $A =
\{0,1\}$) with a given interaction graph, that describes which local functions
depend on which variables. In this paper, we generalise the so-called reduction
of coding functions in order to eliminate variables. We then determine the
maximum number of fixed points of a fully reduced coding function, whose
interaction graph has a loop on every vertex. Since the reduction preserves the
number of fixed points, we then apply these ideas and results to obtain four
main results on the linear network coding solvability problem. First, we prove
that non-decreasing coding functions cannot solve any more instances than
routing already does. Second, we show that triangle-free undirected graphs are
linearly solvable if and only if they are solvable by routing. This is the
first classification result for the linear network coding solvability problem.
Third, we exhibit a new class of non-linearly solvable graphs. Fourth, we
determine large classes of strictly linearly solvable graphs.
