Data Burning: Models & Applications

• Data burning refers to burning-based operations on data structures, notably graph burning where a burning sequence and burning number measure network propagation.
• The literature investigates algorithmic challenges including NP-completeness, spanning tree reductions, and heuristic approximations to study burning processes in diverse graph classes.
• Applied usages extend to satellite detection of crop residue burning and write-once file systems that record data irreversibly.

Data burning, in the literature represented here, denotes a cluster of burning-based operations on data-bearing structures rather than a single standardized term. Its dominant formal usage is graph burning, a deterministic discrete-time process in which one new fire source may be selected in each round while existing fires spread to adjacent vertices; the associated burning number measures how quickly influence, contagion, information, or resources can cover a network (Bonato, 2020). Closely related work studies hypergraph, random, adversarial, and product-graph variants. Separate applied usages concern the detection of crop residue burning from satellite imagery (Walker et al., 2022) and the recording of data to write-once media through append-only file-system transactions (Garfinkel et al., 2020).

1. Formal model and canonical invariants

In graph burning, all vertices are initially unburned. At each round, one unburned vertex is selected and burned, and fire from previously burned vertices spreads to their neighbours. A burning sequence of length $k$ is a sequence $(x_1,\dots,x_k)$ such that

$$N_{k-1}[x_1]\cup N_{k-2}[x_2]\cup \cdots \cup N_0[x_k]=V(G),$$

where $N_d[v]$ is the set of vertices within distance $d$ from $v$ (Bonato et al., 2015). An equivalent sphere-packing formulation is

$$b(G)=\min\left\{k:\exists v_1,\dots,v_k\in V(G),\;V(G)=\bigcup_{i=1}^k B(v_i,k-i)\right\},$$

with $B(v,r)$ the ball of radius $r$ around $v$ (Peca-Medlin, 27 Feb 2026).

The burning number $(x_1,\dots,x_k)$0 is bounded by standard distance parameters. For a connected graph of diameter $(x_1,\dots,x_k)$1 and radius $(x_1,\dots,x_k)$2,

$(x_1,\dots,x_k)$3

and for paths and cycles,

$(x_1,\dots,x_k)$4

(Guillaume et al., 2 Jun 2026). The path formula is central because it is sharp for the conjectured extremal behaviour of connected graphs.

A structurally important reduction states that

$(x_1,\dots,x_k)$5

so burning can be analyzed via spanning trees without changing the optimum (Bonato, 2020). This reduction explains why tree subclasses remain central even when the target graph class is much broader.

2. Conjecture, hardness, and structural characterization

The central open statement is the Burning Number Conjecture: every connected graph $(x_1,\dots,x_k)$6 of order $(x_1,\dots,x_k)$7 satisfies

$(x_1,\dots,x_k)$8

The conjecture is sharp for paths and has been verified for several structured families, but it remains open in general (Bonato, 2020).

A common misconception is that burning becomes algorithmically easy on sparse or acyclic inputs. In fact, the decision problem is NP-complete for trees of maximum degree three, spider graphs, and path-forests (Bonato et al., 2015). Hardness persists on several dense or geometrically defined classes: interval graphs, permutation graphs, and disk graphs are NP-complete for optimal burning (Gupta, 2020), and connected cubic graphs together with connected proper interval graphs remain NP-complete as well (Antony et al., 2024). The broader complexity landscape includes APX-hardness and W[2]-completeness parameterized by $(x_1,\dots,x_k)$9 (Bonato, 2020).

At the small-burning-number end, structural characterizations are unusually sharp. A graph $$N_{k-1}[x_1]\cup N_{k-2}[x_2]\cup \cdots \cup N_0[x_k]=V(G),$$0 of order $$N_{k-1}[x_1]\cup N_{k-2}[x_2]\cup \cdots \cup N_0[x_k]=V(G),$$1 satisfies

$$N_{k-1}[x_1]\cup N_{k-2}[x_2]\cup \cdots \cup N_0[x_k]=V(G),$$2

where $$N_{k-1}[x_1]\cup N_{k-2}[x_2]\cup \cdots \cup N_0[x_k]=V(G),$$3 is the maximum degree (Guillaume et al., 2 Jun 2026). This yields immediate classifications for several graph classes. Connected split graphs satisfy $$N_{k-1}[x_1]\cup N_{k-2}[x_2]\cup \cdots \cup N_0[x_k]=V(G),$$4, and connected $$N_{k-1}[x_1]\cup N_{k-2}[x_2]\cup \cdots \cup N_0[x_k]=V(G),$$5-free graphs satisfy $$N_{k-1}[x_1]\cup N_{k-2}[x_2]\cup \cdots \cup N_0[x_k]=V(G),$$6 (Guillaume et al., 2 Jun 2026).

For induced-path restrictions, connected $$N_{k-1}[x_1]\cup N_{k-2}[x_2]\cup \cdots \cup N_0[x_k]=V(G),$$7-free graphs obey

$$N_{k-1}[x_1]\cup N_{k-2}[x_2]\cup \cdots \cup N_0[x_k]=V(G),$$8

and the bound is tight up to an additive constant $$N_{k-1}[x_1]\cup N_{k-2}[x_2]\cup \cdots \cup N_0[x_k]=V(G),$$9 (Antony et al., 2024). This provides a clean interface between hereditary structure and burning-time control.

3. Dense, tree-like, and product families

Algorithmic progress is especially strong on dense and tree-like graphs. For a graph on $N_d[v]$0 vertices with minimum degree $N_d[v]$1, one algorithm burns the graph in at most

$N_d[v]$2

rounds; for $N_d[v]$3, this is a constant number of rounds (Kamali et al., 2019). The same line of work establishes the conjectured $N_d[v]$4 bound for graphs of minimum degree at least $N_d[v]$5 (Kamali et al., 2019).

For graphs with bounded path-length $N_d[v]$6 and diameter $N_d[v]$7,

$N_d[v]$8

which gives a $N_d[v]$9-approximation for bounded path-length families (Kamali et al., 2019). For bounded tree-length $d$0, an approximation algorithm achieves

$d$1

hence a $d$2-approximation when $d$3 is fixed (Kamali et al., 2019). Interval and chordal graphs appear naturally in this regime.

Grid-like families exhibit more delicate asymptotics. For fence graphs $d$4, explicit improved bounds were derived; in the special case $d$5, the lower bound is $d$6 and the upper bound is $d$7 (Bonato et al., 2019). The significance is not merely asymptotic order $d$8, but the demonstration that the multiplicative constants in the fence regime are substantially larger than earlier extrapolations suggested.

Product constructions have become another major source of exact formulas. For comb graphs $d$9, the Burning Number Conjecture holds for all parameter regimes (Peca-Medlin, 27 Feb 2026). In the spine-dominant regime $v$0,

$v$1

while in general

$v$2

(Peca-Medlin, 27 Feb 2026). On strong products of paths $v$3, burning reduces to a geometric tiling problem with radial propagation, and sharp lower bounds are obtained through an Euler-Maclaurin analysis of sums of odd $v$4th powers (Ambrose et al., 24 Sep 2025).

Highly symmetric families also admit exact formulas. For instance,

$v$5

for even $v$6, and exact formulas are known for $v$7, $v$8, and $v$9 (Fitzpatrick et al., 2017). These results tie burning-time growth directly to neighbourhood expansion in regular vertex-transitive graphs.

4. Hypergraph, stochastic, adversarial, and game-theoretic extensions

Hypergraph burning changes the propagation rule. In one formulation, a vertex $$b(G)=\min\left\{k:\exists v_1,\dots,v_k\in V(G),\;V(G)=\bigcup_{i=1}^k B(v_i,k-i)\right\},$$0 burns in round $$b(G)=\min\left\{k:\exists v_1,\dots,v_k\in V(G),\;V(G)=\bigcup_{i=1}^k B(v_i,k-i)\right\},$$1 if there exists a non-singleton hyperedge $$b(G)=\min\left\{k:\exists v_1,\dots,v_k\in V(G),\;V(G)=\bigcup_{i=1}^k B(v_i,k-i)\right\},$$2 containing $$b(G)=\min\left\{k:\exists v_1,\dots,v_k\in V(G),\;V(G)=\bigcup_{i=1}^k B(v_i,k-i)\right\},$$3 such that all other members of $$b(G)=\min\left\{k:\exists v_1,\dots,v_k\in V(G),\;V(G)=\bigcup_{i=1}^k B(v_i,k-i)\right\},$$4 were burned by round $$b(G)=\min\left\{k:\exists v_1,\dots,v_k\in V(G),\;V(G)=\bigcup_{i=1}^k B(v_i,k-i)\right\},$$5 (Burgess et al., 2024). The lazy variant burns only an initial set and then relies solely on propagation. Unlike graphs, arbitrary hypergraphs do not satisfy an analogue of the sublinear $$b(G)=\min\left\{k:\exists v_1,\dots,v_k\in V(G),\;V(G)=\bigcup_{i=1}^k B(v_i,k-i)\right\},$$6 bound: even linear $$b(G)=\min\left\{k:\exists v_1,\dots,v_k\in V(G),\;V(G)=\bigcup_{i=1}^k B(v_i,k-i)\right\},$$7-uniform loose paths can have

$$b(G)=\min\left\{k:\exists v_1,\dots,v_k\in V(G),\;V(G)=\bigcup_{i=1}^k B(v_i,k-i)\right\},$$8

so $$b(G)=\min\left\{k:\exists v_1,\dots,v_k\in V(G),\;V(G)=\bigcup_{i=1}^k B(v_i,k-i)\right\},$$9 and $B(v,r)$0 can be linear in $B(v,r)$1 (Burgess et al., 2024).

A proportion-based variant introduces a threshold $B(v,r)$2: if at least $B(v,r)$3 vertices of a hyperedge $B(v,r)$4 are on fire, then the entire hyperedge burns in the next round (Burgess et al., 2024). This yields burning distributions that are piecewise constant in $B(v,r)$5, with monotonicity

$B(v,r)$6

(Burgess et al., 2024).

Lazy hypergraph burning admits exact combinatorial characterizations. If $B(v,r)$7 is the maximum cardinality of a $B(v,r)$8-matching in the incidence graph, then

$B(v,r)$9

and for the dual hypergraph,

$r$0

(Bonato et al., 2024). The same work proves equivalence between lazy burning and zero forcing on the incidence graph and establishes NP-completeness for computing an upper bound on the lazy burning number (Bonato et al., 2024).

Random and adversarial models change the source-selection mechanism. On the discrete Euclidean torus $r$1, if each new source is chosen uniformly from the unburned vertices, then the random burning number satisfies

$r$2

where $r$3 is the explosion time of a generalized Blasius equation; in dimension $r$4, $r$5 (Blanc et al., 2 Sep 2025). In the adversarial Builder–Arsonist game on growing graphs, if Builder adds $r$6 vertices at round $r$7, then Arsonist wins for $r$8 and Builder wins for $r$9 (Gunderson et al., 2022). Strong path products also support cooling and liminal burning variants, including the threshold parameter $v$0 (Ambrose et al., 24 Sep 2025).

5. Approximation, heuristics, and empirical methodology

Since exact burning is intractable on many classes, approximation and heuristic methods are central. General-graph approximation factors reported in the literature include factor $v$1 for arbitrary graphs, factor $v$2 for trees, and factor $v$3 for path-forests (Bonato, 2020). For square grids, a $v$4-approximation algorithm is known, together with a lower bound of order $v$5 for an $v$6 grid (Gupta et al., 2020).

The first general heuristics for connected graphs were evaluated on theta graphs, DIMACS, BHOSLIB, and random graphs with fixed distance to cluster (Farokh et al., 2020). Six heuristics were proposed: Ctr-Half-dist, Ctr-Far-dist, Rnd-Half-dist, Rnd-Far-dist, DFS-path, and D-BFS-path. On 78 DIMACS graphs, the heuristics achieved optimal burning numbers of length $v$7 in 71 cases; on 36 BHOSLIB graphs, all heuristics computed optimal sequences of length $v$8 (Farokh et al., 2020). On 2000 random theta-graphs, the heuristics matched the theoretical upper bound in $v$9 of instances and were within $(x_1,\dots,x_k)$00 in $(x_1,\dots,x_k)$01, with DFS-path achieving optimality in $(x_1,\dots,x_k)$02 (Farokh et al., 2020).

Parameter-based upper bounds can also guide heuristic design. If $(x_1,\dots,x_k)$03 is a cluster graph, then

$(x_1,\dots,x_k)$04

which improves the earlier $(x_1,\dots,x_k)$05 bound for distance to cluster and yields $(x_1,\dots,x_k)$06 for the constructed family studied experimentally (Farokh et al., 2020).

A practical blueprint for empirical work on social networks is to compute $(x_1,\dots,x_k)$07, $(x_1,\dots,x_k)$08, diameter, radius, density, and the degree sequence; apply bounds such as $(x_1,\dots,x_k)$09; test for high-degree structure via $(x_1,\dots,x_k)$10; and then use class-specific results or brute-force verification on moderate instances (Guillaume et al., 2 Jun 2026). This suggests that burning studies increasingly combine structural recognition, parameter bounds, and computational experimentation rather than relying on a single analytic technique.

6. Separate applied usages: satellite detection and write-once storage

Outside graph theory, burning appears in distinct data-centric meanings. In remote sensing, the problem is to detect crop residue burning from satellite imagery. A study in Punjab, India used ground-verified data from 3206 rice plots, including 441 burned and 240 unburned plots, together with 3 m PlanetScope imagery and Sentinel-2 imagery (Walker et al., 2022). After separability analysis of bands and burn indices, a Random Forest model was built and achieved an overall model accuracy of $(x_1,\dots,x_k)$11-percent (Walker et al., 2022). The work identified rapid signal decay, cloud and haze gaps, and the spectral ambiguity of shadows, bare soil, and tilled fields as central technical constraints.

A separate storage-systems usage concerns burning data to write-once media. A file-system standard for such media, including digital compact disks, was proposed as a full-featured and high-performance alternative to conventional file systems on multiple-write media (Garfinkel et al., 2020). The detailed design is append-only: changes are committed as transactions, previous versions remain available through version chains, and structures such as the Directory List and End-Of-Transaction block allow the file system to work with any operating system and a variety of physical media (Garfinkel et al., 2020). In this context, burning refers to irreversible recording rather than diffusion.

Taken together, these literatures show that “data burning” is not a single formal object. The dominant mathematical tradition is graph burning and its generalizations, where burning quantifies propagation speed on combinatorial structures. Separate applied traditions use the same language for physical burning observed in data and for data written irreversibly to storage media. A plausible implication is that the unifying theme is not the substrate—graphs, images, or disks—but the study of irreversible spread, recording, or state transition under strong structural constraints.