Is the B Train an Express Train?

Subtitle: An esoteric mathematical exploration of “local” vs. “express” in the New York City subway system for people with too much free time.

In order to answer the question of the B Train, we must first tackle a deeper philosophical question: what does it mean to be an express train?

Perhaps “local” vs “express” is in the eye of the B-holder. An entertainment executive zipping along from their Park Slope abode to their office in Rockefeller Center would find the B train to be sufficiently express. Whereas an Upper West Sider making their way from their gastroenterologist at Mount Sinai West to their dermatologist on 86th and CPW might find themselves at a crawl on a local B train, regretting not trusting their initial instinct to take the 1. And what would both of these people have to say about a train that can’t even be bothered to run at night or on the weekends?

If you now believe that a train’s local- or express-ness is a matter of subjective opinion, you would be wrong. It is actually a matter of irrefutable mathematical fact. To understand this, we introduce the following key idea: a train can only be express relative to another train.

Specifically, train X is express relative to train Y if train Y makes a series of consecutive stops a_1, a_2, …, a_n, and train X stops at a_1 and then, immediately after, a_n. Conversely, if train X is express relative to train Y, train Y is local relative to train X.

A visualization of the mathematical definition above: if you have a length of track like this, this means the E and the F are express relative to the M and the R.

Now that we have a definition for what it means to be express, we can answer the question posed at the beginning. The B train is express relative to the F, M, and Q trains, and local relative to the A and D trains. However, we can can take this concept further.

A truly disturbed individual might be inclined to construct a directed graph to visualize the local-express relationships among trains in the system. If such an individual were to construct such a graph, it would look like this:

This graph accounts for weekday daytime service. For the sake of simplicity, we exclude trains which are not part of a local-express relationship (G, L, S, 7) and diamond express variants (6, 7, F).

From this graph we can make a few interesting observations:

• There are five trains that are purely express (A, D, Z, 4, 5), seven trains that are purely local (C, J, M, R, W, 1, 6), and seven trains that are both local and express (B, E, F, N, Q, 2, 3). The G, L, 7, and all three shuttles are neither local nor express.
• The D train is the “most express” train as it is express relative to the greatest number of trains (B, F, M, R).
• The R train is the “most local” train as it is local relative to the greatest number of trains (D, E, F, N, Q). I imagine those of you who regularly ride the R train reached the same conclusion intuitively.
• The D is express relative to the B which is express relative to the F which is express relative to the M. This makes the BDFM line “well-ordered” in terms of express-ness. It is the only “well-ordered” line with more than two trains.
• If we were to include nighttime service, that would make the D express relative to the A, the F express relative to the E, and the 3 express relative to the 2. Making the D express relative to the A means that the D is recursively express relative to all the trains in its graph.
• One nice property that emerges from the graph is that there are no cycles (i.e., no situation where two trains are both local and express relative to each other). However, the presence of this property relies on a subtlety in the definition of stations. The Q is express relative to the N from 42nd St to 57th St, with the N stopping at 49th St in between. However, one might also say the N is express relative to the Q from Canal St to Atlantic Av, with the Q stopping at DeKalb Av in between. To avoid this perilous situation, we consider separate lines (per the MTA definition of a line) stopping at the same station to be separate stations. In this case, it means that the Atlantic Av stop on the Q (Brighton Line) and the Atlantic Av stop on the N (Fourth Avenue Line) are separate, and as such, the N is not express relative to the Q (even though it will get you to the Barclays Center slightly faster).