A graph is a collection of vertices and edges, where vertices represent entities and an edge between two vertices implies that the corresponding entities are related. A path in a graph is a sequence of consecutive edges, i.e., the terminal vertex of an edge is the same as the initial vertex in the next edge in the path. A path that begins and ends at the same vertex is called a circuit or cycle. A Euler circuit in a graph is a simple circuit containing every edge of the graph. A Euler path in a graph is a simple path containing every edge of the graph.

The problem of whether a graph has a Euler circuit was first informally stated around the end of the 18th century in the town Königsberg, Prussia - now called Kaliningrad and part of the Russian Federation - regarding whether it was possible to start at some location in the town, travel across all the seven bridges connecting banks of the Pregel river once without crossing any bridge twice, and return to the starting point. A solution to this problem was suggested and published by the Swiss mathematician Leonard Euler in 1736 giving rise to the field of Graph Theory. Euler circuits and paths - which are studied in the context of various university courses in computer science and other disciplines - can be used to address many practical problems in communication, transportation or even biological networks where a circuit or path traversing every edge of the corresponding graph structure is required.

Serious games, i.e., games focusing on educational purposes rather than just entertainment, can be highly effective in facilitating active learning and deeper understanding of complex concepts. Through interactive gameplay, well-designed serious games maintain engagement, encourage critical thinking and improve problem-solving skills.

In this work, we present a serious game intended to support students in their study regarding Euler circuits and paths in graphs. According to the plot of the game players are explorers who are presented with complexes of islands and connections among them and they are requested to discover circuits or paths crossing every connection exactly once, if feasible. Players can select to play a single round or a layered variant. In single rounds, players define their desired difficulty level in the sense that they provide the necessary parameters for the generation of random graphs corresponding to complexes of islands and connections among them. The layered variant includes three levels - beginner, regular, advanced - where graphs, i.e., island complexes, are automatically generated by the application. In both cases, before asked to draw an Euler circuit or path, players are first asked to determine whether such circuits or paths exist in each instance. Correctly answering these questions involves appropriate background in graph theory while drawing Euler circuits or paths involves knowledge of corresponding algorithmic solutions from the relevant literature. Our serious game enables players to test and strengthen their theoretical knowledge by practically applying it in instances of the game.

Our game is currently available in English and Greek and it is freely accessible via any web browser. It has been built using Godot, a cross-platform, free and open-source game engine which has gained attention mainly due to its built-in tools that facilitate the creation of complex user interfaces.

Keywords: Serious game, island complexes, explorers, Euler circuit, Euler path, Fleury algorithm, Godot game engine.