Biregularity in Sidorenko's Conjecture

Sidorenko's conjecture says that the minimum density of a bigraph $G$ in a bigraphon $W$ of a given edge density is attained when $W$ is a constant function. We reduce Sidorenko's conjecture to the case when the target bigraphon $W$ is biregular. With this biregularity result and some ideas of its proof, we also obtain simple proofs in the asymmetric setting of several results previously shown in the symmetric setting. Furthermore, we also show that bigraphs that have a special type of tree decomposition, called reflective tree decomposition, satisfy Sidorenko's conjecture. This both unifies and generalizes the notions of strong tree decompositions and $N$-decompositions from the literature.