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

Reduction and Fixed Points of Boolean Networks and Linear Network Coding Solvability