Max Capture Time (CR-Ordinal) in Cops and Robbers

• Maximum Capture Time (CR-Ordinal) is an ordinal-valued measure that extends capture time by quantifying finite yet unbounded game durations in cops and robbers scenarios on graphs.
• It employs a recursive ordinal relation to evaluate each finite play’s duration, demonstrating that while every game finishes in finite moves, no global finite bound exists.
• A novel infinite graph construction using a grid and coordinate axes shows that the maximum capture time equals ω, thereby refuting earlier conjectures.

Searching arXiv for the main CR-ordinal paper and closely related cops-and-robbers ordinal work to ground the article. Search query: (Flídr et al., 29 Jul 2025) cops robbers CR-ordinal Bonato Gordinowicz Hahn Maximum capture time, also called the CR-ordinal, is an ordinal-valued refinement of capture time in the classical cops and robbers game on a graph. In the formulation used for infinite graphs, it records the supremal capture complexity over cop starting positions, with pairwise capture times themselves defined by a transfinite relation on vertices. The central development in the recent literature is the construction of an infinite cop-win graph in which every fixed pair of starting positions yields capture in some finite number of moves, but no uniform finite bound exists over all pairs. Consequently, the graph has maximum capture time exactly $\omega$, the first infinite ordinal, thereby disproving a conjecture of Bonato, Gordinowicz, and Hahn that $\omega$ is not a CR-ordinal (Flídr et al., 29 Jul 2025).

1. Game-theoretic setting

The underlying game is the standard cops and robbers pursuit game on a fixed connected graph $G$. The cop chooses a starting vertex, then the robber chooses his starting vertex. The players then move alternately, with the cop moving first, and on each turn a player may either move to an adjacent vertex or stay put. The cop wins if he lands on the robber’s vertex; if the robber is never captured, then the robber wins. A graph is called cop-win if the cop has a winning strategy (Flídr et al., 29 Jul 2025).

For finite graphs, the paper recalls a structural notion that underlies the classical theory. A vertex $y$ dominates $x$ if

$N[x]\subseteq N[y],$

where $N[x]$ denotes the closed neighborhood of $x$. Finite graphs are cop-win if and only if they are constructible, meaning that they can be built from a one-vertex graph by repeatedly adding dominated vertices. For infinite graphs, the paper states that there is no such complete characterization (Flídr et al., 29 Jul 2025).

Within this framework, capture time is not merely a winning-versus-losing dichotomy. It measures how long the cop must play, under optimal play, before capture is forced. In finite graphs this quantity is integer-valued, but on infinite graphs the natural extension is ordinal-valued. That extension is what gives rise to the CR-ordinal (Flídr et al., 29 Jul 2025).

2. Ordinal formulation of capture time

For finite cop-win graphs, the paper defines several related quantities. If the robber starts at $u$ and the cop at $v$, and the robber moves first, then $\omega$0 is the number of turns needed for capture. The worst-case capture time for a fixed cop start $\omega$1 is

$\omega$2

the usual capture time of the graph is

$\omega$3

and the maximum capture time is

$\omega$4

This last quantity is the one generalized to the ordinal setting (Flídr et al., 29 Jul 2025).

For infinite graphs, these values need not be finite integers. The paper therefore introduces an ordinal framework. If $\omega$5 has cardinality $\omega$6, it sets

$\omega$7

and defines relations $\omega$8 recursively on vertices by

$\omega$9

and, for $G$0,

$G$1

The pairwise capture time is then

$G$2

and the maximum capture time is again obtained from

$G$3

An ordinal that arises in this way is called a CR-ordinal (Flídr et al., 29 Jul 2025).

For finite values, this ordinal definition agrees with the usual game length. The case of primary interest in the 2025 construction is the limit situation in which every $G$4 is finite, yet the set of such values is unbounded in $G$5. In that case the supremum is

$G$6

so the graph has first-infinite maximum capture time even though no individual game lasts infinitely long (Flídr et al., 29 Jul 2025).

3. The graph $G$7 with CR-ordinal $G$8

The graph constructed in the paper is an infinite graph $G$9 with vertex set

$y$0

Its edges are defined by

$y$1

The paper describes the resulting geometry as follows: all vertices on the $y$2-axis are mutually adjacent, all vertices on the $y$3-axis are mutually adjacent, and off the axes, a vertex is adjacent to vertices strictly “up-left” or “down-right” from it (Flídr et al., 29 Jul 2025).

The main theorem asserts that $y$4 is cop-win and satisfies a stronger finiteness property: for any two starting vertices $y$5, there exists a finite $y$6 such that if the cop starts at $y$7 and the robber at $y$8, then the cop can guarantee capture in at most $y$9 turns. However, there is no single finite $x$0 that works for all starting positions. Equivalently, every fixed initial position pair has finite capture time, but the set of capture times is not uniformly bounded. Therefore,

$x$1

The theorem is explicitly stated as showing that $x$2 is a CR-ordinal (Flídr et al., 29 Jul 2025).

This graph is notable because it isolates a purely ordinal phenomenon. The obstruction is not that some starting pairs require genuinely transfinite play. Rather, every game is finitely terminating, but the finite termination bounds diverge along appropriately chosen starting positions. The ordinal $x$3 therefore appears as a supremum of finite capture times, not as a pairwise capture time that is already infinite (Flídr et al., 29 Jul 2025).

4. Mechanism of the cop’s winning strategy

The proof that $x$4 is cop-win proceeds by exploiting the special role of the coordinate axes. If the robber is at a vertex of the form $x$5 or $x$6 with $x$7, then he is trapped very quickly: the cop can move to some $x$8 with $x$9, and then any legal robber move either stays on or reaches the $N[x]\subseteq N[y],$0-axis, where capture is immediate, or moves up-left, which is also adjacent to $N[x]\subseteq N[y],$1. By symmetry, the same statement holds near the $N[x]\subseteq N[y],$2-axis (Flídr et al., 29 Jul 2025).

The main forcing argument concerns robber positions $N[x]\subseteq N[y],$3 with $N[x]\subseteq N[y],$4 when it is the cop’s turn. The cop moves to

$N[x]\subseteq N[y],$5

which forces the robber, if he wishes to avoid immediate capture, to move down-right. This decreases the robber’s $N[x]\subseteq N[y],$6-coordinate. Repeating this maneuver forces the robber to reach the $N[x]\subseteq N[y],$7-axis in at most $N[x]\subseteq N[y],$8 steps, after which capture occurs within at most two more moves (Flídr et al., 29 Jul 2025).

The significance of this strategy is that it uses the axes as “highways” of complete adjacency while using the off-axis comparability relation to impose monotone descent on one robber coordinate. Since the decreasing coordinate is a natural number, it cannot decrease forever. The paper therefore concludes that $N[x]\subseteq N[y],$9 is cop-win (Flídr et al., 29 Jul 2025).

The paper also notes a side result: $N[x]$0 is even constructible. That observation is relevant because constructibility is the finite-graph hallmark of cop-win structure, yet here it appears in an infinite setting in which the ordinal behavior is more subtle (Flídr et al., 29 Jul 2025).

5. Finite but unbounded capture times

The argument that $N[x]$1 requires more than proving that the cop always wins. It must also show that no global finite bound exists on capture time. To do this, the paper proves that for every $N[x]$2, one can choose starting positions sufficiently far out in the grid so that the robber survives at least $N[x]$3 turns. Roughly, if both players begin with coordinates at least $N[x]$4, then the robber can move so as to decrease one coordinate only slowly, never getting forced into immediate capture before reaching an axis or a near-axis position (Flídr et al., 29 Jul 2025).

Two conclusions follow. First, every starting pair has some finite capture time. Second, the collection of these times is unbounded in $N[x]$5. Hence

$N[x]$6

and therefore

$N[x]$7

The paper then states this directly in the ordinal framework: if $N[x]$8 is the constructed graph, then $N[x]$9; in other words, $x$0 is a CR-ordinal (Flídr et al., 29 Jul 2025).

This phenomenon clarifies an important point about ordinal capture time. A graph may be cop-win without possessing a uniform finite capture bound. In such a case, the distinction between “the cop always wins” and “the cop wins within bounded time” becomes mathematically consequential. The CR-ordinal records that distinction exactly (Flídr et al., 29 Jul 2025).

6. Conjecture, significance, and terminological scope

The construction settles a specific conjectural question. Bonato, Gordinowicz, and Hahn had shown that certain ordinals are CR-ordinals, namely

$x$1

and they conjectured that these are all the CR-ordinals. In particular, they conjectured that

$x$2

is not a CR-ordinal. The graph $x$3 disproves that conjecture, since it satisfies $x$4 (Flídr et al., 29 Jul 2025).

The paper also raises a further question: whether there exists a cop-win graph with $x$5 and some pair $x$6 with $x$7. This separates two ways in which $x$8 might arise: as the supremum of finite pairwise values, or as an actual pairwise capture time (Flídr et al., 29 Jul 2025).

A terminological clarification is useful because the expression “ordinal” appears in unrelated areas. In “Detecting Change-Points in Time Series by Maximum Mean Discrepancy of Ordinal Pattern Distributions,” the method uses ordinal pattern distributions and MMD for change-point localization, but the paper does not name its performance notion CR-Ordinal explicitly; its relevant notion is change-point localization of $x$9 by maximizing MMD or CMMD (Sinn et al., 2012). Likewise, “Provably Minimum-Length Conformal Prediction Sets for Ordinal Classification” does not use the terms “maximum capture time,” “CR-Ordinal,” “capture rate,” or “ordinal capture regions” explicitly; its closest analogue is a minimum-length conformal prediction interval for ordinal labels, where “capture” is whether the true label lies inside a contiguous ordinal interval (Zhang et al., 20 Nov 2025). These usages are conceptually adjacent only at the level of vocabulary. In graph pursuit, by contrast, CR-ordinal has a specific technical meaning tied to ordinal-valued capture time in cops and robbers (Flídr et al., 29 Jul 2025).