Studying minimal separators in graph theory is important for both practical and theoretical reasons. Some graphs have polynomially many minimal separators, while others have exponentially many. Recently, there has been increasing interest in understanding which graph structures lead to exponentially many minimal separators. One such structure is a k -creature. It has been observed that graphs containing a k -creature as an induced subgraph have exponentially many minimal separators. Because of this, it was initially conjectured that forbidding k -creatures would be sufficient to characterize graph classes with only a polynomial number of minimal separators. However, this conjecture was disproven by Gartland and Lokshtanov, who introduced a graph class known as k -twisted ladders. Although these graphs are free of k -creatures, they still have exponentially many minimal separators. This counterexample motivated the introduction of a new graph structure called the k -critter, which generalizes the k -twisted ladder. This paper aims to clarify the fundamental differences between k -creatures and k -critters by analyzing their associated lattice structures. We focus on two representative graph types: k -ladders, which exemplify k -creatures, and k -twisted ladders, which represent the k -critter structure. By comparing the lattices formed by the minimal separators of each graph, we demonstrate that, despite their similar graph structures, their lattice structures differ significantly. Our results show that lattice-theoretical methods provide useful insights for studying minimal a, b -separators.

We explore the feasibility of early introduction to automata theory through gamification. We designed a puzzle game that players can answer correctly if they understand the fundamental concepts of automata theory. In our investigation, 90 children played the game, and their actions were recorded in play logs. An analysis of the play logs shows that approximately 60% of the children achieved correct-answer rates of at least 70%, which suggests that primary and lower secondary school students can understand the fundamental concepts of automata theory. Meanwhile, our analysis shows that most of them do not fully understand automata theory, but some of them have a good understanding of the concept.