Semi-transitivity of directed split graphs generated by morphisms

Iamthong, Kittitat and Kitaev, Sergey (2023) Semi-transitivity of directed split graphs generated by morphisms. Journal of Combinatorics, 14 (1). ISSN 2156-3527 (

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A directed graph is semi-transitive if and only if it is acyclic and for any directed path u1 → u2 → ··· → ut, t ≥ 2, either there is no edge from u1 to ut or all edges ui → uj exist for 1 ≤ i < j ≤ t. In this paper, we study semi-transitivity of families of directed split graphs obtained by iterations of morphisms applied to the adjacency matrices and giving in the limit infinite directed split graphs. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. We fully classify semi-transitive infinite directed split graphs when a morphism in question can involve any n×m matrices over {−1,0,1} with a single natural condition.