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The problem of finding graph structure of functions commuting with a givenfunction in terms of their functional graphs is considered. Structure offunctional graphs of commuting functions is described. The problem is reducedto describing graph homomorphisms of weakly connected components of functionalgraphs. Four subcases with finite sets are considered: permutations commutingwith permutation, permutations commuting with a function, functions commutingwith a permutation and functions commuting with a function. For finite sets thenumber of functions commuting with a given one and functions with extremalproperties are found. Results for finite sets are generalized to the case ofarbitrary sets where there are additional types of functional graph components.

Graph structure of commuting functions