Multi-Eulerian Tours of Directed Graphs

Not every graph has an Eulerian tour. But every finite, strongly connected graph has a multi-Eulerian tour, which we define as a closed path that uses each directed edge at least once, and uses edges e and f the same number of times whenever tail(e)=tail(f). This definition leads to a simple generalization of the BEST Theorem. We then show that the minimal length of a multi-Eulerian tour is bounded in terms of the Pham index, a measure of 'Eulerianness'.

Recovering Drawing Trajectory

Graph theory has many applications in solving real-life problems. However, the application of Eulerian graphs and Eulerian tours/trails seems to be comparatively limited. In this chapter, an application of graph theory in handwriting recognition is presented. There are a lot of studies regarding handwriting recognition. Most of these methods deal with either offline or online handwriting recognition. However, the discussed approaches in this chapter are distinct in the manner that they aim to extract the trajectory of writing so as to equip the offline handwritten image with temporal information. When the trajectory of writing is known, it can be possible to utilize online recognition methods which are more reliable. These trajectory extracting methods are based on Eulerian trails in semi-Eulerian graphs. Semi-Eulerian graphs are graphs with at most two odd vertices. Eulerian trail is a trail in which every edge is traversed exactly once. The methods can be helpful in recognition of single-stroke handwritten images. Relying on the minimum energy law, the methods try to find the smoothest trajectory of writing which contribute to the recognition process.

Chinese Postman Problem

One of the very popular applications of the graph theory in real world problems is related to the concept of Eulerian tours and trails introduced in Eulerian trail and tours chapter. There are many problems in which users should serve all the connections (edges in a graph, streets of a city, pipelines of a network and etc.) between nodes. In chapter 7 of this book, the existence of such trails and tours in graphs were discussed, and appropriate algorithms were introduced to find Eulerian trails and tour. But in the case a graph does not have such a tour or trail, it’s important to traverse some edges more than once, and this is what usually happens in real world applications. M.K. Kwan in 1962 was the first who introduced this problem as the Chinese postman problem (CPP). The question was that, given a postal zone with a number of streets that must be served by a postal carrier, how can one develop a tour that covers every street in the zone and brings the postman back to his or her point of origin, having traveled the minimum possible distance (Wang et al., 2008)? In this chapter, the Chinese postman problem is discussed, and different variations of it are introduced. Then the very early form of the CPP in which the graph is undirected is explained in more detail.