A King in every two consecutive tournaments
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We think of a tournament T=([n], E) as a communication network where in each round of communication processor P_i sends its information to P_j, for every directed edge ij ∈ E(T). By Landau's theorem (1953) there is a King in T, i.e., a processor whose initial input reaches every other processor in two rounds or less. Namely, a processor P_ν such that after two rounds of communication along T's edges, the initial information of P_ν reaches all other processors. Here we consider a more general scenario where an adversary selects an arbitrary series of tournaments T_1, T_2,..., so that in each round s=1, 2, ..., communication is governed by the corresponding tournament T_s. We prove that for every series of tournaments that the adversary selects, it is still true that after two rounds of communication, the initial input of at least one processor reaches everyone. Concretely, we show that for every two tournaments T_1, T_2 there is a vertex in [n] that can reach all vertices via (i) A step in T_1, or (ii) A step in T_2 or (iii) A step in T_1 followed by a step in T_2.
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