Semi-transitivity of directed split graphs generated by morphisms

TL;DR

This paper investigates the semi-transitivity property of infinite directed split graphs generated by morphisms on adjacency matrices, providing a complete classification under certain matrix conditions.

Contribution

It introduces a classification of semi-transitive infinite directed split graphs generated by morphisms on adjacency matrices with specific constraints.

Findings

Full classification of semi-transitive infinite directed split graphs.

Analysis of morphisms involving matrices over {-1,0,1}.

Conditions under which semi-transitivity holds in generated graphs.

Abstract

A directed graph is semi-transitive if and only if it is acyclic and for any directed path $u_1\rightarrow u_2\rightarrow \cdots \rightarrow u_t$, $t \geq 2$, either there is no edge from $u_1$ to $u_t$ or all edges $u_i\rightarrow u_j$ exist for $1 \leq i < j \leq t$. 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 $n\times m$ matrices over $\{-1,0,1\}$ with a single natural condition.