Tron, a combinatorial Game on abstract Graphs

The movie Tron from 1982 inspired the computer game Tron [4]. The game is played in a rectangle by two light cycles or motorbikes, which try to cut each other off, so that one, eventually has to hit a wall or a light ray. We consider a natural abstraction of the game, which we define as follows: Given an undirected graph G, two opponents play in turns. The first player (Alice) begins by picking a start-vertex of G and the second player (Bob) proceeds by picking a different start-vertex for himself. Now Alice and Bob take turns, by moving to an adjacent vertex from their respective previous one in each step. While doing that it is forbidden to reuse a vertex, which was already traversed by either player. The game ends when both players cannot move anymore. The competitor who traversed more vertices wins. Tron can be pictured with two snakes, which eat up pieces of a tray of cake, with the restriction that each snake only eats adjacent pieces and starves if there is no adjacent piece left for her. We assume that both contestants have complete information at all times.

Bodlaender is the one who first introduced the game to the science community, and according to him Marinus Veldhorst proposed to study Tron.

Bodlaender showed PSPACE-completeness for directed graphs with and without given start positions [1]. Later, Bodlaender and Kloks showed that there are fast algorithms for Tron on trees [2] and NP-hardness and co-NP-hardness for undirected graphs.

We have two kind of results. On the one hand we investigated by how much Alice or Bob can win at most. It turns out, that both players can gather all the vertices except a constant number in particular graphs. This results still holds for k-connected graphs. For planar graphs, we achieve a weaker, but similar result. We also investigated the computational complexity question. we showed PSPACE-completeness for Tron played on undirected graphs both when starting positions are given and when they are not given.

Many proofs require some tedious case analysis. We therefore believe that thinking about the cases before reading all the details will facilitate the process of understanding. To simplify matters, we neglected constants whenever possible.

The aim of this section is to show some basic characteristics and introduce some notation, so that the reader has the opportunity to become familiar with the game.

Definition. Let G be a graph, and Alice and Bob play one game of Tron on G. Then we denote with #A the number of vertices Alice traversed and with #B the number of vertices Bob traversed on G. The outcome of the game is #B/#A. We say Bob wins iff #B > #A, Alice wins iff #A > #B and otherwise we call the game a tie. We say Bob plays rationally, if his strategy maximizes the outcome and we say Alice plays rationally if her strategy minimizes the outcome. We further assume that Alice and Bob play always rational.

Here we differ slightly from [1], where Alice loses if both players receive the same amount of vertices. We introduce this technical nuance, because it makes more sense in regard of the extremal question and is not relevant for the complexity question. Now when you play a few games of Tron on a “random” graph, you will observe that you will usually end up in a tie or you will find that one of the players made a mistake during the game. So a natural first question to ask is if Alice or Bob can win by more than one at all.

Example 1 (two paths). Let G be a graph which consists of two paths of length n. On the one hand, Alice could start close to the middle of one of the paths, then Bob starts at the beginning of the other path, and thus wins. On the other hand if Alice tries to start closer to the end of a path Bob will cut her off by starting next to her on the longer side of her path. The optimal solution lies somewhere in between and a bit of arithmetic reveals that for the optimal choice #B/#A tends to 2/( √ 5 -1) as n tends to infinity.

And what about Alice? We will modify our graph above by adding a super-vertex v adjacent to every vertex of G. Now when Alice starts there the first vertex on G will be taken by Bob and Alice will take the second vertex on G. So we see that the roles of Alice and Bob have interchanged.

Lemma 1 (Super-vertex). Let G be a graph where (#B/ #A) G > 1 and F be the graph we obtain by adding a super-vertex v adjacent to every vertex of

So lemma 1 simplifies matters. Once we have found a good graph for Bob we have automatically found a good graph for Alice. But the other direction holds as well. Let G be a graph where Alice wins and let us say she starts at vertex v. Delete vertex v from G to attain H. Now the situation in Alice’s first move in H is the same as Bob’s first move in G. And Bob’s first move in H includes the options Alice had in her second move in G.

Lemma 2. Let G be a graph where (#B/ #A) G < 1 and H be the graph we obtain by deleting the vertex where Alice starts. It follows that

Note that the starting

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