Arc-disjoint Strong Spanning Subdigraphs of Semicomplete Compositions
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A strong arc decomposition of a digraph D=(V,A) is a decomposition of its arc set A into two disjoint subsets A_1 and A_2 such that both of the spanning subdigraphs D_1=(V,A_1) and D_2=(V,A_2) are strong. Let T be a digraph with t vertices u_1,... , u_t and let H_1,... H_t be digraphs such that H_i has vertices u_i,j_i, 1< j_i< n_i. Then the composition Q=T[H_1,... , H_t] is a digraph with vertex set ∪_i=1^t V(H_i)={u_i,j_i| 1< i< t, 1< j_i< n_i} and arc set (∪^t_i=1A(H_i) ) ∪( ∪_u_iu_p∈ A(T){u_ij_iu_pq_p| 1< j_i< n_i, 1< q_p< n_p}). We obtain a characterization of digraph compositions Q=T[H_1,... H_t] which have a strong arc decomposition when T is a semicomplete digraph and each H_i is an arbitrary digraph. Our characterization generalizes a characterization by Bang-Jensen and Yeo (2003) of semicomplete digraphs with a strong arc decomposition and solves an open problem by Sun, Gutin and Ai (2018) on strong arc decompositions of digraph compositions Q=T[H_1,... , H_t] in which T is semicomplete and each H_i is arbitrary. Our proofs are constructive and imply the existence of a polynomial algorithm for constructing a decomposition of a digraph Q=T[H_1,... , H_t], with T semicomplete, whenever such a decomposition exists.
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