## 1 Introduction
### Related Previous Works  
\"Overtaking stations\" are stations on double-track lines with sidings on each of the two sides of the main line; and one or two, or more crossovers at the two throats of the overtaking station [3] (Figures 3 and 4). Almost all the small overtaking stations on double-track lines have the same track layouts everywhere (sometimes with fewer crossovers); they can be found on high-speed lines of countries such as Japan, France, Spain, etc. [4], as well as on classic lines (in fact, the track layouts of Figures 3 and 4 are considered standards in [4] (p. 155)). Observe that we are not considering in the track layout diagrams the frequently used dead-end sidings included for safety reasons; this is because they do not add anything to the connectivity of the station, which is the goal of this article. Figure 3: Track layout diagram of a small overtaking station on a double-track line. It has one sliding on each of the sides of the main line and two crossovers at the two throats of the station. For example, in the Spanish high-speed network, the following stations have exactly this track layout: Bujaraloz, Ballobar, LEspluga, El Prat (Spanish high-speed line Madrid–Barcelona [4] (p. 62)); Puente Genil (Spanish high-speed line Cordoba–Malaga [4] (p. 66)); Requena-Utiel (Spanish high-speed line Madrid–Valencia [4] (p. 68)). Figure 2: A light signal in a main station. The author is a mathematician and computer scientist with a long experience in the design, development, and implementation of software for railways. Some of these works [5] have been developed in cooperation with or commissioned by the \"Fundacion de los Ferrocarriles Espanoles\" (Spanish Railway Foundation) [6]. Regarding track layout design [7], he implemented in the computer algebra system (CAS) [8,9]_Maple_ (_Maple_ is a trademark of _Waterloo Maple Inc._) [10, 11, 12, 13] a small topology-independent package [14] based on the use of cycles in an undirected graph [15]; which could find compatible routes in overtaking stations on double-track lines (of any topology). Let us observe that the compatibility of routes has to be very carefully checked during the design of a railway interlocking system. A collateral result found was the existence of an alternative track layout for small overtaking stations on double-track lines (Figure 5), with advantages and disadvantages over the standard track layout. The disadvantage is that only half of the tracks are reachable from each throat of the station. The advantage is that section 0 allows to maintain a bidirectional use of the station on a degraded scenario where the tracks on one side of the main line are unavailable. The next goal was to design, develop, and implement an ad hoc package focused on comparing different track layouts of overtaking stations on a double-track line (of any topology) [16]. The software chosen was also _Maple_. The key idea was to compare alternative track layouts and to calculate the number of pairs of non-conflicting routes (one in each direction) in each case; simulating the ones that could be simultaneously authorized by a railway interlocking system. Figure 4: Track layout diagram of another small overtaking station on a double-track line (it has two sidings on each of the sides of the main line and two crossovers at the two throats of the station). For example, the station Cuenca Fernando Zobel (on the Spanish high-speed line Madrid–Valencia) has exactly this track layout [4] (p. 68). Fukushima station (on the Japanese Tohoku Shinkansen line) is very close to this one; the only difference is that it has only one crossover at one of the throats of the station [4] (p. 36). Similarly, Camp de Tarragona station (on the Spanish high-speed line Madrid–Barcelona [4] (p. 62)) has three sidings on each of the sides of the main line; however, it has crossovers only on one throat of the station. Figure 5: An alternative track layout diagram for a small overtaking station on a double-track line.