Broom in Graph Theory, Engineering & Imaging

• Broom is a graph-theoretic construct combining a path with concentrated leaves, serving as a design pattern in random-walk and extremal problems.
• In software and systems engineering, BROOM refers to methodologies and tools that streamline real-time automotive control and model tidying.
• Push-broom imaging and 'broom star' in astronomy illustrate the visual metaphor of a handle with a brush, applied in remote sensing and historical comet observations.

BROOM is a recurrent term in several research literatures, but it does not denote a single canonical object. In graph theory, it usually names a tree built from a path together with a cluster of pendent leaves concentrated near one end; this family appears in random-walk extremal problems, forbidden-induced-subgraph theory, extremal combinatorics, approximation algorithms, and chromatic symmetric functions. In software and systems engineering, broom and BROOM designate tools or methodologies for tidying model outputs, developing real-time automotive controllers, and performing blind component separation in microwave astronomy. In remote sensing, push-broom refers to a line-scanning sensor geometry, and in historical East Asian astronomy broom star denotes a cometary appearance (Beveridge et al., 4 Aug 2025, Robinson, 2014, Keto et al., 2024, Fuchs et al., 2014, Carones et al., 15 Apr 2026, Neuhaeuser et al., 2020).

1. Graph-theoretic definitions and notational variants

The graph-theoretic literature uses several non-identical broom constructions. Their common motif is a path-like “handle” together with a concentrated set of leaves or a subdivided star arm. This notational overload is substantive rather than merely cosmetic: different variants are tailored to random walks, forbidden induced subgraphs, spanning-tree existence, or extremal counting.

Variant	Construction	Source
$B_{n,d}$	$n$-vertex tree from a path of length $d$ with $n-d$ additional leaves at one endpoint	(Beveridge et al., 4 Aug 2025)
$B(\ell,s)$	Path $P_\ell$ with $s$ new leaves attached to the penultimate vertex $v_{\ell-1}$	(Gerbner, 2024)
$B_{m,n}$	Path of length $n$ with $n$0 extra leaves attached at the unique vertex at distance $n$1 from one end	(Wang et al., 2021)
$n$2	Path $n$3 plus $n$4 new vertices adjacent only to one endpoint $n$5	(Bacsó et al., 2018)
$n$6-broom	Graph obtained from $n$7 by subdividing exactly one edge once	(Liu et al., 2021)
Broom of length $n$8	$n$9-edge path whose non-path vertices are adjacent exactly to one end	(Scott et al., 2018)
Spanning broom	Spanning spider obtained by joining the center of a star to one endpoint of a path	(Kim et al., 2024)
$d$0	Path of length $d$1, $d$2 leaves at one end, plus an extra pendant leaf attached at $d$3	(Rahimi et al., 9 Dec 2025)

Two further extensions are prominent. A $d$4-multibroom is obtained by identifying the handles of brooms of lengths $d$5, producing a tree with several broom-arms meeting at one common vertex (Scott et al., 2018). In bipartite graph mining, a $d$6-broom is defined not as a forbidden tree but as the unique spanning tree of a $d$7 under a fixed vertex order, with $d$8 edges “evenly distributed” across the two sides (Chen et al., 15 May 2025).

This suggests that “broom” functions less as a single graph family than as a path-plus-brush design pattern whose exact formalization depends on the problem being studied.

2. Random walks, meeting time, and spanning existence

In the random-walk setting, the broom graph is an extremal tree for meeting time. For a random walk on a tree $d$9, with hitting time $n-d$0 and stationary distribution $n-d$1, the meeting time is

$n-d$2

For fixed order $n-d$3 and diameter $n-d$4, this quantity is maximized by the broom graph. The meeting time is minimized by the balanced double broom graph, or a slight variant, depending on the relative parities of $n-d$5 and $n-d$6 (Beveridge et al., 4 Aug 2025).

For the broom $n-d$7, the maximum of the scaled functional

$n-d$8

is attained at a leaf $n-d$9, and the paper gives the closed forms

$B(\ell,s)$0

and

$B(\ell,s)$1

The proof strategy is structural: maximizing targets are leaves, relocating leaves toward the maximizing leaf strictly increases the meeting time, and repeated reattachment of off-geodesic branches eventually “broomifies” the tree (Beveridge et al., 4 Aug 2025).

A distinct extremal line concerns spanning containment under degree conditions. Chen, Ferrara, Hu, Jacobson, and Liu proved that every connected graph of order $B(\ell,s)$2 with

$B(\ell,s)$3

contains a spanning broom. Kim, Kostochka, and Luo established the Ore-type corollary that every connected $B(\ell,s)$4-vertex graph with $B(\ell,s)$5 and

$B(\ell,s)$6

also contains a spanning broom; the Ore-type result is obtained by first embedding a spanning jellyfish and then deleting an edge to obtain a broom (Kim et al., 2024).

Taken together, these results place the broom in two complementary roles: as an extremizer of a random-walk functional on trees of fixed diameter, and as a spanning structure forced by sharp degree or degree-sum thresholds.

3. Forbidden brooms and chromatic structure

For induced-subgraph theory, broom exclusion yields several polynomial $B(\ell,s)$7-binding results. A $B(\ell,s)$8-broom is obtained from $B(\ell,s)$9 by subdividing one edge once. For every fixed integer $P_\ell$0, the class of $P_\ell$1-broom-free graphs is polynomially $P_\ell$2-bounded; specifically, there exists a polynomial $P_\ell$3 such that $P_\ell$4. When $P_\ell$5, every chair-free graph satisfies

$P_\ell$6

For $P_\ell$7, every graph with no induced $P_\ell$8-broom and no copy of $P_\ell$9 satisfies

$s$0

An explicit general form is

$s$1

and in the $s$2-free case one may take

$s$3

(Liu et al., 2021).

Multibrooms extend this program. Scott and Seymour proved that every $s$4-multibroom satisfies the Gyárfás–Sumner conjecture, unifying earlier results for the uniform $s$5 and $s$6 cases (Scott et al., 2018). In a different direction, Rahimi and Mojdeh studied the hereditary class $s$7 of $s$8-free graphs. If

$s$9

then

$v_{\ell-1}$0

For the subclass additionally excluding $v_{\ell-1}$1, the chromatic number is linearly bounded by the clique number, and for each $v_{\ell-1}$2 the subclass $v_{\ell-1}$3-free graphs admits a linear $v_{\ell-1}$4-binding function $v_{\ell-1}$5 (Rahimi et al., 9 Dec 2025).

These results show that broom-type trees are useful forbidden configurations precisely because they are structured enough to admit Ramsey-type, separator-type, or template-based control of chromatic growth, while still being rich enough to capture nontrivial induced-subgraph behavior.

4. Extremal, algorithmic, and enumerative roles

The broom also appears as a proxy object in algorithm design and extremal counting. In bipartite graphs, the CBC framework for approximate $v_{\ell-1}$6-biclique counting introduces the $v_{\ell-1}$7-broom as a special spanning tree of the biclique. After coloring the graph, all brooms are counted exactly by a dynamic program

$v_{\ell-1}$8

with total complexity $v_{\ell-1}$9 time and $B_{m,n}$0 space. Sampling these brooms yields an unbiased estimator $B_{m,n}$1 for the true biclique count $B_{m,n}$2, with Hoeffding-type bound

$B_{m,n}$3

Empirically, the method achieves up to $B_{m,n}$4 reduction in estimation error and up to $B_{m,n}$5 speedup on nine real-world bipartite networks (Chen et al., 15 May 2025).

In forbidden-subgraph extremal theory, Gerbner determined $B_{m,n}$6 and $B_{m,n}$7 for $B_{m,n}$8, any $B_{m,n}$9, and sufficiently large $n$0. Writing $n$1, the unique extremal graphs are $n$2 when $n$3 is even, $n$4 when $n$5 is odd with either $n$6 or $n$7, and $n$8 when $n$9 (Gerbner, 2024).

For optimization on broom-free graphs, there is a polynomial-space algorithm which, on any $n$00-vertex $n$01-free graph $n$02, returns in time

$n$03

an independent set of size at least

$n$04

The proof combines degree-threshold branching, a path-growing/small-separator lemma, and a “poor neighborhood” case in which a small part of the graph is solved exactly (Bacsó et al., 2018).

In algebraic combinatorics, the broom graph $n$05 has a sharply constrained positivity profile. Wang, Wang, and Wang proved that $n$06 is $n$07-positive if and only if $n$08. They also showed that the only Schur-positive brooms beyond the trivial path cases are

$n$09

Their methods use Orellana–Scott triple-deletion and a rim-hook–tabloid formula for Schur coefficients (Wang et al., 2021).

A plausible implication is that broom structures are unusually effective as intermediate objects: they are simple enough to count, sample, or analyze exactly, yet rich enough to preserve the combinatorial difficulty of the ambient problem.

5. Software packages and engineering methodologies named broom or BROOM

In statistics, the R package broom is designed to convert statistical analysis objects into tidy data frames. It adopts the tidy-data framework summarized by three rules: “Each variable forms one column. Each observation forms one row. Each type of observational unit forms one table.” The package defines exactly three S3 generics: tidy(model, ...), augment(model, data=...), and glance(model, ...), corresponding respectively to component level, observation level, and model level. Each returns a data.frame without rownames, making model outputs easier to recombine, filter, reshape, and visualize with tidyverse tools (Robinson, 2014).

In automotive real-time software development, BROOM means “BMW-ROOM.” It is a pragmatic version of ROOM adapted to ETAS’s ASCET SD for tool-assisted development of car software components. The method organizes development in two nested loops: a Product Development Loop and a System Development Loop with the phases “Model Requirements Definition,” “Model Development,” and “Implementation & Test.” BROOM is scenario-driven, prototype-driven, and actor-based: systems are decomposed into actors with ports, attributes, subactors, and state machines, and ASCET’s code-generation and experimentation features are used to validate emerging models “on button press” (Fuchs et al., 2014).

In microwave astrophysics, BROOM is a Python package for model-independent analysis of microwave data. It implements Internal Linear Combination methods and a Generalized ILC framework for blind reconstruction of coherent emission components with unknown covariance properties. The package is pure Python and installable via $n$24 It supports simulation, component separation, residual estimation, and angular power-spectrum estimation. Validation is reported for a full-sky satellite mission and a partial-sky ground-based experiment. In the LiteBIRD-like configuration, “CMB reconstruction unbiased” and “foreground residuals suppressed by factor $n$10” are reported; in the SO-SATs-like case, $n$11-mode spectra are recovered to $n$12 accuracy over $n$13, and GILC-recovered foreground spectral power matches inputs within $n$14 at high $n$15 (Carones et al., 15 Apr 2026).

These software and methodology uses are nominal rather than structural: unlike the graph-theoretic broom, they are brand names or acronyms, but each preserves the broader idea of organizing heterogeneous elements into a regular, manipulable form.

6. Push-broom imaging and “broom star” astronomy

In Earth observation, push-broom denotes a sensor geometry in which multi-spectral imagery is acquired by sweeping linear detector arrays across-track while the satellite moves along-track. PlanetScope SuperDove is described with typical parameters

$n$16

Because the spectral strips are read out sequentially, a moving object appears at slightly different along-track positions in different bands. With

$n$17

band-to-band displacements can be converted into velocity estimates. Keto and Watters analyzed eight aircraft tracks over Dallas–Fort Worth International Airport, inferred

$n$18

and reported an RMSE of PLC versus ADS-B of $n$19 ($n$20); $n$21 error bars envelop the one-to-one line, confirming accuracy to within $n$22 at the $n$23 confidence level (Keto et al., 2024).

In historical astronomy, broom star translates the Chinese term hui xing. Neuhäuser and collaborators show that the AD 668 and AD 891 records often misidentified in modern speculative counterpart studies are best interpreted as comet sightings. The AD 668 reports describe a broom star at Wuche, Bi, and Mao lasting 19 nights, while the AD 891 materials combine Japanese and Chinese reports of an object with motion through Santai, Taiwei, Dajiao, and Tianshi and a duration of at least 54 days. Tail descriptions, motion across asterisms, and the absence of stationarity satisfy comet criteria and not nova or supernova criteria (Neuhaeuser et al., 2020).

Outside graph theory, then, the broom metaphor is visual and geometric. In push-broom imaging it refers to a scanning strip architecture; in historical astronomy it refers to a tailed appearance. This suggests that across disciplines the term retains a common image—a handle-like axis together with a brush-like extension—even when the formal objects are entirely different.