Broom Graphs: Structure & Applications

• Broom graphs are trees characterized by a long handle and a brush of pendant edges, offering a versatile template across graph theory.
• They are constructed by subdividing a star or extending a path, with applications in chromatic bounds, extremal problems, and spectral optimization.
• Special cases and variants (t-brooms, double-brooms, minimal brooms) reveal nuanced roles in induced subgraph theory and algorithmic graph analysis.

A broom graph is a tree organized around a handle and a brush: one arm is a path, while the remaining arms are pendant edges attached at one endpoint or near one endpoint of that path. The term is used in several closely related but not identical senses across graph theory, extremal combinatorics, graph coloring, random walks, and spectral graph theory. In the ordinary tree-theoretic usage, a broom is obtained either from a path by adding leaves to one end, or from a star by subdividing one edge; specialized literatures also study $t$-brooms, double-brooms, spanning brooms, minimal brooms, and short brooms (Liu et al., 2021, Gerbner, 2024, Nguyen et al., 9 May 2026, Kim et al., 2024).

1. Core graph-theoretic form

Several standard definitions describe the same general shape: one distinguished arm is longer than the others, and one branching vertex carries the brush.

Notation	Exact definition	Typical setting
$t$-broom	obtained from $K_{1,t+1}$ by subdividing an edge once	induced-subgraph coloring (Liu et al., 2021)
$(k,\ell)$-broom	obtained from $K_{1,\ell+1}$ by subdividing an edge exactly $k-2$ times	Ramsey-type $\chi$-bounds (Nguyen et al., 9 May 2026)
$(t,k)$-broom	obtained from $K_{1,t+1}$ by subdividing an edge $k$ times	generalized broom notation (Rahimi et al., 9 Dec 2025)
$t$0	obtained from an $t$1-vertex path by adding $t$2 new leaves connected to a penultimate vertex	forbidden-subgraph extremal theory (Gerbner, 2024)
$t$3	a path $t$4 with $t$5 additional degree-one vertices adjacent to one endpoint	MIS approximation (Bacsó et al., 2018)
$t$6	obtained from a handle $t$7 by adding $t$8 pendant edges to an end vertex	rainbow Turán theory (Byrne et al., 22 Feb 2025)

The paper on generalized broom obstructions states that “a broom is a tree consisting of a high-degree center together with one distinguished arm that is longer than the others” (Rahimi et al., 9 Dec 2025). The paper on spanning structures formulates the same geometry as “a spanning spider obtained by joining the center of a star to an endpoint of a path,” and immediately rephrases this as “a broom is a spider with all but at most one leg of length $t$9” (Kim et al., 2024). This suggests a common editorial synthesis: a broom graph is the one-branching-vertex tree obtained by attaching a star-like brush to one end, or near one end, of a path.

Two equivalent construction principles recur. One starts from a star and subdivides a chosen edge; the other starts from a path and adds pendant leaves to one endpoint or to a penultimate vertex. In both formulations the result is a tree with one branching point and a single distinguished handle (Liu et al., 2021, Gerbner, 2024).

2. Small instances, special cases, and related families

Small parameter values recover standard graphs. For the $K_{1,t+1}$0-broom of induced-subgraph theory, a $K_{1,t+1}$1-broom is $K_{1,t+1}$2, so $K_{1,t+1}$3-broom-free graphs are exactly $K_{1,t+1}$4-free graphs; a $K_{1,t+1}$5-broom is the chair or fork graph (Liu et al., 2021). In the $K_{1,t+1}$6-broom notation, $K_{1,t+1}$7-broom $K_{1,t+1}$8, $K_{1,t+1}$9-broom is $(k,\ell)$0, and $(k,\ell)$1-broom is again the fork or chair (Nguyen et al., 9 May 2026). In the $(k,\ell)$2 notation, $(k,\ell)$3, so forbidding a broom interpolates between forbidding a path and forbidding a one-branching tree (Gerbner, 2024).

Some papers use parameters tied to diameter or to a star–path decomposition. The random-walk paper defines $(k,\ell)$4 as a path $(k,\ell)$5 with leaves $(k,\ell)$6 incident with $(k,\ell)$7; the handle is $(k,\ell)$8 and the bristles are $(k,\ell)$9. In that notation, $K_{1,\ell+1}$0 and $K_{1,\ell+1}$1 (Beveridge et al., 4 Aug 2025). The network-centrality paper defines $K_{1,\ell+1}$2 by connecting the path graph $K_{1,\ell+1}$3 by a bridge to the center of the star graph $K_{1,\ell+1}$4; with its conventions, $K_{1,\ell+1}$5 has $K_{1,\ell+1}$6 vertices and $K_{1,\ell+1}$7 edges, so it is a tree (Dangalchev, 2023).

A closely related two-ended family is the double-broom. The reconstruction paper defines $K_{1,\ell+1}$8 as the tree obtained from a $K_{1,\ell+1}$9-vertex path by appending $k-2$0 leaf neighbors at one end and $k-2$1 leaf neighbors at the other end; when $k-2$2, this is the usual double-star (Ma et al., 2016). Double-brooms recur in random-walk extremal problems as balanced or near-balanced analogues of the one-sided broom (Beveridge et al., 4 Aug 2025, Beveridge et al., 28 Oct 2025).

3. Forbidden induced subgraphs and $k-2$3-boundedness

Brooms are central in the theory of $k-2$4-bounded graph classes. For $k-2$5-brooms, the main result is that for graphs $k-2$6 without induced $k-2$7-brooms,

$k-2$8

For $k-2$9, the bound is strengthened to

$\chi$0

and for $\chi$1 and $\chi$2-free graphs,

$\chi$3

The same paper notes that a $\chi$4-broom is the chair or fork, that $\chi$5-broom-free graphs were already known to be $\chi$6-bounded, and that the new contribution is polynomial control with substantially sharper bounds in the chair-free case (Liu et al., 2021).

A Ramsey-type extension replaces ordinary polynomial $\chi$7-bounds by linear bounds in Ramsey numbers. For every $\chi$8, $\chi$9, $(t,k)$0, and $(t,k)$1, there exists $(t,k)$2 such that every $(t,k)$3-free graph $(t,k)$4 satisfies

$(t,k)$5

More generally, if every component of a forest $(t,k)$6 is a broom, then the $(t,k)$7-free class is $(t,k)$8-bounded by a linear function of $(t,k)$9 (Nguyen et al., 9 May 2026).

Brooms also appear in Gyárfás–Sumner theory through multibrooms. A broom of length $K_{1,t+1}$0 is defined there as a tree obtained from a $K_{1,t+1}$1-edge path with ends $K_{1,t+1}$2 by adding some number of leaves adjacent to $K_{1,t+1}$3, with $K_{1,t+1}$4 called the handle. A $K_{1,t+1}$5-multibroom is obtained by identifying the handles of brooms of lengths $K_{1,t+1}$6. The main theorem proves that every $K_{1,t+1}$7-multibroom satisfies the Gyárfás–Sumner conjecture (Scott et al., 2018).

A later refinement introduces generalized brooms $K_{1,t+1}$8, described as a “generalized broom with an additional leaf,” inside the hereditary class $K_{1,t+1}$9 of $k$0-free and $k$1-flag-free graphs. There the authors prove that if $k$2 does not contain $k$3 as a subgraph and is $k$4-free, then $k$5 for a polynomial $k$6; for fixed $k$7 and $k$8, if $k$9 is $t$00-free and $t$01-free, then there exists a linear function $t$02 such that $t$03 (Rahimi et al., 9 Dec 2025).

4. Extremal, enumerative, and algorithmic settings

Forbidden broom subgraphs support exact extremal results for degree statistics and star counts. For the broom $t$04 obtained from an $t$05-vertex path by adding $t$06 new leaves connected to a penultimate vertex, the degree-power paper determines $t$07 and $t$08 for every $t$09, all $t$10, and sufficiently large $t$11. Writing

$t$12

the extremal graphs are $t$13 when $t$14 is even, $t$15 when $t$16 is odd and $t$17 or when $t$18, and $t$19 when $t$20 and $t$21 (Gerbner, 2024).

In algorithmic graph theory, broom exclusion yields a subexponential approximation scheme for MIS but not an exact algorithm of the same type as in $t$22-free graphs. For fixed integers $t$23, one can find a $t$24-approximation to Maximum Independent Set on an $t$25-vertex $t$26-free graph $t$27 in time

$t$28

The proof exploits the observation that if a vertex $t$29 has an independent $t$30-set in its neighborhood, then an induced $t$31 starting at $t$32 would create an induced $t$33 (Bacsó et al., 2018).

Rainbow Turán theory isolates a particularly rigid short-handle family. The broom $t$34 is the graph obtained from a handle $t$35 by adding $t$36 pendant edges to an end vertex; the paper concentrates on $t$37, described as “stars with a single edge subdivided twice” (Byrne et al., 22 Feb 2025). The asymptotic results are highly sensitive to divisibility: $t$38

$t$39

$t$40

and, when $t$41,

$t$42

The same paper proves exact asymptotics $t$43 when $t$44, and a matching lower bound when $t$45 (Byrne et al., 22 Feb 2025).

5. Random walks, centrality, and Steklov extremality

For simple random walk on trees, the broom is an extremal shape for hitting-time functionals under fixed order and diameter. If

$t$46

then for fixed order $t$47 and diameter $t$48, $t$49 is achieved uniquely by the broom $t$50. The maximizing target is the handle tip $t$51, and the explicit value is

$t$52

In the same framework, the balanced double broom graph, or a slight variant depending on parity, minimizes the corresponding meeting time among trees of order $t$53 and diameter $t$54 (Beveridge et al., 4 Aug 2025).

For the best meeting time

$t$55

the extremal picture is reversed: among trees of order $t$56 and diameter $t$57, the maximizing graph is the balanced double broom $t$58, while the minimizing graph is the balanced lever. The paper nonetheless retains the one-sided broom $t$59 as a formal family and as a decomposition piece, since splitting a balanced double broom at a barycenter produces broom components (Beveridge et al., 28 Oct 2025).

Distance-based graph invariants also admit closed broom formulas. Under Dangalchev’s exponential closeness,

$t$60

the broom graph $t$61 formed by joining a path $t$62 to the center of a star $t$63 by a bridge satisfies

$t$64

and its line graph satisfies

$t$65

These formulas are obtained from bridge decompositions on $t$66, $t$67, and their line graphs (Dangalchev, 2023).

In spectral graph theory, brooms are the basic minimizers for mixed Steklov problems. The paper on combinatorial Steklov eigenvalues defines $t$68 by attaching a Dirichlet boundary edge of length $t$69 to one endpoint of a path of length $t$70, and $t$71 boundary vertices to the other endpoint. Its first Steklov eigenvalue is

$t$72

The minimizers of higher Steklov eigenvalues are then assembled from minimal brooms: if $t$73, the minimum of the $t$74 Steklov eigenvalue is attained by a star with each arm a minimal broom; if $t$75, it is attained by a regular comb with each tooth a minimal broom (Yu et al., 2022).

6. Specialized extensions and terminology drift

Several papers use “broom” for structures that preserve the handle–brush motif but serve more specialized purposes. In spanning-tree theory, a broom is a target spanning structure rather than a fixed forbidden graph. The jellyfish paper studies connected $t$76-vertex graphs with

$t$77

and proves that such a graph contains a spanning broom; it also recalls the earlier result that every connected graph of order $t$78 with

$t$79

contains a spanning broom (Kim et al., 2024).

In edge-coloring, a short broom is a color-structured sequence

$t$80

inside a $t$81-critical graph, with the condition that each color $t$82 is missing at one of the earlier vertices. The paper proves

$t$83

equivalently: at most one color is missing at more than one vertex of the short broom, and if such a color exists, it is missing at exactly two vertices (Chen et al., 8 Dec 2025).

In approximate biclique counting, a $t$84-broom is not a tree family in the usual hereditary sense but a canonical spanning tree of a $t$85-biclique. The paper defines a $t$86-broom as a special spanning tree with $t$87 edges, counts these objects exactly by a color-ordered dynamic program in

$t$88

and then uses broom counts to build an unbiased estimator for $t$89-biclique counts (Chen et al., 15 May 2025).

Finally, double-brooms remain important in reconstruction theory. For every double-broom $t$90, the degree-associated edge-reconstruction numbers $t$91 and $t$92 are determined completely; the answer is “usually $t$93” for $t$94 and “$t$95” for $t$96, with explicit exceptional families (Ma et al., 2016).

Across these literatures, the most stable feature of the broom graph is not a single notation but a structural template: one branching region, one distinguished handle, and a brush of pendant edges. The term therefore functions both as a precise graph family and as a transferable motif linking induced-subgraph structure, extremal graph theory, edge-coloring, random walks, and spectral optimization.