🤖 AI Summary
This study investigates whether a directed graph admits an edge labeling over a finite alphabet that renders it a completely reachable automaton, and characterizes those graphs—termed fully labeling-robust—for which every possible labeling yields complete reachability. By integrating tools from graph theory, automata theory, and computational complexity, the work provides the first complete structural characterization of directed graphs that can realize completely reachable automata. The main contributions include a polynomial-time algorithm for recognizing such graphs, a proof that the decision problem is NP-complete when the alphabet size is fixed, and a full classification of fully labeling-robust graphs.
📝 Abstract
We determine which digraphs admit an edge labeling by letters from a finite alphabet such that the resulting labeled digraph is a completely reachable automaton. Such digraphs are recognizable in polynomial time; however, the problem becomes NP-complete when the size of the label alphabet is fixed. We also classify the digraphs for which every edge labeling results in a completely reachable automaton.