Localization Game in Graph Theory

• The Localization Game is a pursuit–evasion game on graphs where cops use distance probes to pinpoint an invisible robber, blending adaptive strategy with metric dimension ideas.
• Key parameters such as the localization number and capture time quantify the minimum probes and rounds needed, with established bounds for trees, outerplanar graphs, and hypercubes.
• Recent research extends the game to visibility-limited, directed, and random models, refining bounds and revealing implications for algorithmic graph theory and network localization.

The localization game is a pursuit–evasion game on a graph in which an invisible robber moves between adjacent vertices while a team of cops probes vertices and receives distance information. The principal invariant is the localization number $\zeta(G)$, the minimum number of probes per round that guarantees eventual identification of the robber’s position; later work introduced localization capture time to quantify the number of rounds required under optimal play. The subject is commonly treated as a game-theoretic variant of metric dimension, but its sequential and adaptive structure has produced a distinct theory involving exact values, extremal bounds, probabilistic asymptotics, directed and visibility-limited variants, and recent linear capture-time results for trees and subclasses of outerplanar graphs (Chenoweth et al., 14 Aug 2025).

1. Formal model and basic parameters

In the standard finite undirected setting, the game is played on a finite, connected graph $G=(V,E)$. Two players participate: a team of $k$ cops and one robber. The robber is invisible and omniscient, the cops have global movement, and a round consists of a cops’ move followed by a robber’s move. On the cops’ move, the cops choose a multiset of $k$ vertices $C=\{c_1,\dots,c_k\}$, immediately receiving the distance vector

$(d(c_1,r),\;d(c_2,r),\;\dots,\;d(c_k,r)),$

where $r$ is the robber’s current vertex. The robber then moves along one edge or stays put. The cops win if after some finite round they can deduce the robber’s exact location; otherwise the robber wins by evading forever (Chenoweth et al., 14 Aug 2025).

A vertex is a candidate if its distances to the probed vertices match the reported distances. The game therefore admits a state-space interpretation in which each probe refines the current candidate set, and the robber attempts to keep that set non-singleton across rounds. This viewpoint underlies many later proofs, including “robber-territory” arguments for incidence graphs and inductive shrinking arguments for trees (Bonato et al., 2021).

Three parameters organize most of the theory.

Parameter	Definition
$\zeta(G)$	Minimum $k$ such that $k$ cops have a winning strategy
$G=(V,E)$0	Minimum $G=(V,E)$1 for which the robber can be identified in one round
$G=(V,E)$2	Minimum number of rounds needed by $G=(V,E)$3 cops under optimal play

The relation to metric dimension is immediate: $G=(V,E)$4 is the minimum number of probes needed for one-round identification, so $G=(V,E)$5. The inequality can be strict, and several random and incidence-based families exhibit a substantial gap between sequential localization and one-round resolving power (Bonato et al., 2021).

2. Structural bounds, graph parameters, and complexity

The earliest structural results already separated localization from classical resolving-set phenomena. Seager proved that every finite tree $G=(V,E)$6 satisfies $G=(V,E)$7, with $G=(V,E)$8 exactly when $G=(V,E)$9 does not contain $k$0 as an induced subgraph (Chenoweth et al., 14 Aug 2025). More generally, there is a pathwidth bound

$k$1

and a maximum-degree bound

$k$2

For hypercubes,

$k$3

showing that high Cartesian dimension does not force linear growth in the parameter (Bonato et al., 2021).

The parameter also interacts with sparsity and coloring. If $k$4 has degeneracy $k$5, then $k$6; equivalently, $k$7 implies $k$8. Consequently,

$k$9

The degeneracy bound is tight, and the same work proved that every outerplanar graph satisfies $k$0 (Bonato et al., 2018).

A recurrent misconception is that small treewidth should force small localization number. The established upper bound is in terms of pathwidth, not treewidth, and this distinction matters: there exist planar graphs of treewidth $k$1 with unbounded $k$2 (Bosek et al., 2017). The algorithmic side is correspondingly difficult. Determining whether $k$3 is NP-hard even for graphs with diameter at most $k$4 (Bosek et al., 2017).

These facts place localization in an unusual position among graph-search parameters. It is controlled by path-like decompositions, sensitive to local branching and degeneracy, but not bounded by treewidth alone.

3. Capture time and the linear localization program

Localization capture time measures not merely whether the cops can win, but how fast they can do so. For any $k$5, the $k$6-localization capture time $k$7 is the minimum number of rounds needed for $k$8 cops, playing optimally to minimize time, to locate the robber when the robber responds so as to maximize this time. When $k$9, one writes $C=\{c_1,\dots,c_k\}$0 (Chenoweth et al., 14 Aug 2025).

This led to the Localization Capture Time Conjecture (LCTC): there exists a constant $C=\{c_1,\dots,c_k\}$1 such that for every connected $C=\{c_1,\dots,c_k\}$2-vertex graph $C=\{c_1,\dots,c_k\}$3,

$C=\{c_1,\dots,c_k\}$4

Earlier work established that trees and interval graphs are well-localizable, and for trees $C=\{c_1,\dots,c_k\}$5 on $C=\{c_1,\dots,c_k\}$6 vertices gave the bounds

$C=\{c_1,\dots,c_k\}$7

(Behague et al., 2021).

Recent results substantially sharpen the tree theory. If a tree $C=\{c_1,\dots,c_k\}$8 does not contain $C=\{c_1,\dots,c_k\}$9, then one cop suffices and

$(d(c_1,r),\;d(c_2,r),\;\dots,\;d(c_k,r)),$0

where $(d(c_1,r),\;d(c_2,r),\;\dots,\;d(c_k,r)),$1 is the number of leaves. If $(d(c_1,r),\;d(c_2,r),\;\dots,\;d(c_k,r)),$2 does contain $(d(c_1,r),\;d(c_2,r),\;\dots,\;d(c_k,r)),$3, then $(d(c_1,r),\;d(c_2,r),\;\dots,\;d(c_k,r)),$4 and

$(d(c_1,r),\;d(c_2,r),\;\dots,\;d(c_k,r)),$5

Hence for any tree $(d(c_1,r),\;d(c_2,r),\;\dots,\;d(c_k,r)),$6 on $(d(c_1,r),\;d(c_2,r),\;\dots,\;d(c_k,r)),$7 vertices,

$(d(c_1,r),\;d(c_2,r),\;\dots,\;d(c_k,r)),$8

The proofs root $(d(c_1,r),\;d(c_2,r),\;\dots,\;d(c_k,r)),$9 at a leaf and maintain a vertex whose descendant-subtree contains the robber, eliminating branches by probing leaves and children of degree-$r$0 vertices, and in the two-cop case showing that every probe removes at least two new leaves, with four removed in the first move (Chenoweth et al., 14 Aug 2025).

Outerplanar graphs now provide a second major capture-time class. If an outerplanar graph $r$1 decomposes into edge-disjoint blocks $r$2, then

$r$3

Hence any outerplanar $r$4-vertex graph with $r$5 satisfies $r$6. If $r$7 is $r$8-connected outerplanar with $r$9 chords, then

$\zeta(G)$0

This proves the LCTC for all outerplanar graphs $\zeta(G)$1 with $\zeta(G)$2 (Chenoweth et al., 14 Aug 2025).

The same paper introduced an abstract coloring-based generalization. Given colorings $\zeta(G)$3, one forms a layered game structure on subsets $\zeta(G)$4 and then a reduced game structure by pruning to subsets lying on maximal parent–child chains down to singletons. If the colorings are all distance-colorings induced by subsets of size at most $\zeta(G)$5, then the height of the reduced game structure equals $\zeta(G)$6. This recasts capture time as a combinatorial problem on colorings and parent–child relations. The open question asks whether there exists a function $\zeta(G)$7 such that every $\zeta(G)$8-vertex graph and every list of colorings yields a game-structure of height at most $\zeta(G)$9; a positive answer with $k$0 would settle the LCTC (Chenoweth et al., 14 Aug 2025).

4. Exact values, extremal families, and probabilistic regimes

Several graph families admit exact values or asymptotically tight formulas. In incidence graphs of designs, the localization number is tightly controlled by the design parameters. If $k$1 is a projective plane of order $k$2, then the incidence graph satisfies

$k$3

For an affine plane of order $k$4,

$k$5

More generally, for a $k$6-BIBD one has $k$7, while for a symmetric BIBD $k$8 one has $k$9. Kneser graphs of diameter $k$0 also admit asymptotic formulas: for fixed even $k$1 and $k$2,

$k$3

and for fixed odd $k$4,

$k$5

These results rely on robber-territory induction, counting, and hypergraph-detection arguments (Bonato et al., 2021).

Cartesian products produce another developed subtheory. For connected graphs $k$6 and $k$7,

$k$8

and

$k$9

where $G=(V,E)$00 is the minimum size of a doubly resolving set of $G=(V,E)$01. On toroidal grids, if $G=(V,E)$02, then

$G=(V,E)$03

(Boshoff et al., 2020).

Random models demonstrate that sequential localization often scales very differently from one-round metric dimension. For dense $G=(V,E)$04 with $G=(V,E)$05, if $G=(V,E)$06 is the unique integer with $G=(V,E)$07 and $G=(V,E)$08, then a.a.s.

$G=(V,E)$09

In particular, whenever $G=(V,E)$10,

$G=(V,E)$11

If $G=(V,E)$12, then $G=(V,E)$13 and

$G=(V,E)$14

In the diameter-two regime with fixed $G=(V,E)$15, one also has

$G=(V,E)$16

a.a.s. (Dudek et al., 2019).

For random geometric graphs $G=(V,E)$17 slightly above connectivity, localization admits four asymptotic regimes. In particular, just above connectivity,

$G=(V,E)$18

up to polylogarithmic factors (Lichev et al., 2021).

5. Directed, visibility-limited, and infinite variants

The localization game has been extended in several orthogonal directions. For digraphs, the distance from $G=(V,E)$19 to $G=(V,E)$20 is the length of a shortest directed path from $G=(V,E)$21 to $G=(V,E)$22, or $G=(V,E)$23 if none exists. If $G=(V,E)$24 are the strongly connected components of a digraph $G=(V,E)$25, then

$G=(V,E)$26

and this bound is sharp. Directed width parameters again control the game:

$G=(V,E)$27

There also exists a family of digraphs of order $G=(V,E)$28 with localization number $G=(V,E)$29. For random tournaments $G=(V,E)$30,

$G=(V,E)$31

and doubly regular tournaments, including Paley tournaments, satisfy the same logarithmic bounds (Bonato et al., 2022).

Visibility-limited models alter the sensing channel rather than the graph class. In the $G=(V,E)$32-visibility localization game, a probe reports exact distance only up to radius $G=(V,E)$33, returning $G=(V,E)$34 otherwise. The corresponding invariant $G=(V,E)$35 is nonincreasing in $G=(V,E)$36, with $G=(V,E)$37. The theory differs sharply from the classical finite-tree case: for every $G=(V,E)$38 there exists a finite tree $G=(V,E)$39 with $G=(V,E)$40, yet for any tree $G=(V,E)$41 and any $G=(V,E)$42 there exists a subdivision $G=(V,E)$43 with $G=(V,E)$44 (Bonato et al., 2023). The one-visibility case is particularly developed: $G=(V,E)$45 for $G=(V,E)$46-minor-free graphs of order $G=(V,E)$47, and $G=(V,E)$48 Cartesian grids achieve this order up to additive constants (Bonato et al., 2023).

Infinite graphs require new phenomena. For locally finite graphs, $G=(V,E)$49 may be finite or $G=(V,E)$50. In marked contrast to finite trees, for every positive integer $G=(V,E)$51 and also for $G=(V,E)$52, there exists a locally finite tree $G=(V,E)$53 with $G=(V,E)$54. These examples have uncountably many ends. By contrast, locally finite trees with finitely or countably many ends satisfy $G=(V,E)$55. As in the finite setting, any locally finite graph contains a subdivision with localization number $G=(V,E)$56 (Bonato et al., 2024).

6. Open problems, scope, and terminological overlap

Several open problems currently organize the subject. The localization capture-time program asks for a full resolution of the LCTC. For outerplanar graphs, the remaining case is $G=(V,E)$57. For trees, the linear bound has already been improved to $G=(V,E)$58, and the next step is to identify tighter lower-bound examples. The coloring-structure formulation isolates a still more general question: whether the height of the game structure is bounded by $G=(V,E)$59 for all $G=(V,E)$60-vertex graphs and all lists of colorings. Structural extensions from trees and outerplanar graphs to wider families, including planar and minor-closed classes, remain explicitly proposed directions (Chenoweth et al., 14 Aug 2025).

The phrase “localization game” also appears in other research areas, but with different formal content. In multi-agent SLAM, GTP-SLAM formulates a “Localization Game” as a potential game in which the ego player and non-ego players optimize trajectories, controls, and landmarks, and the iterative best-response procedure converges to an open-loop Nash equilibrium (Chiu et al., 2022). In sensor network localization, non-anchor nodes have been modeled as players in a non-convex potential game whose Nash equilibrium corresponds to the localization solution, with centralized and distributed NE-seeking algorithms (Xu et al., 2024, Xu et al., 2024). In underwater sensor networks, localization has been cast as a single-leader multi-follower Stackelberg game for topology control and energy-efficient ranging (Yuan et al., 2018). These usages share the word “localization” and a game-theoretic vocabulary, but they are distinct from the cops-and-robber distance-probing game on graphs.

Within graph theory proper, the localization game now spans exact finite combinatorics, probabilistic methods, decomposition-based upper bounds, temporal complexity, and several nonclassical sensing models. The most robust theme is that adaptive probing is substantially more powerful than one-round resolution, yet still delicate enough that branching structure, graph products, width parameters, and motion constraints all remain visible in the final bounds.