Double Broom in Graph Theory

• Double broom is a tree structure defined by a path with leaf sets attached at two distinguished vertices, serving as a stable motif in various graph theory analyses.
• It is significant in areas such as reconstruction theory, chromatic symmetric functions, and extremal Wiener index studies, with variants like balanced and strong double brooms.
• Its structure optimizes key metrics in random-walk analyses, including mixing and meeting times, and influences edge ideal depth and degree-associated reconstruction parameters.

Searching arXiv for recent and relevant papers on double broom graphs and related extremal/structural results. In graph theory, a double broom is a tree obtained by attaching leaves to two distinguished vertices on a path, but the exact formalization depends on the problem domain. In reconstruction theory it is the tree $D_{m,n,p}$ obtained from a $p$-vertex path by appending $m$ leaves to one end and $n$ leaves to the other (Ma et al., 2016). In extremal distance theory it is a tree with exactly two broom vertices, so that all leaves are adjacent to one of those two vertices (Borovićanin et al., 23 Jul 2025). In chromatic symmetric function work it appears as the family $\mathrm{br}'(l,p,l')$, built by joining two broom-like end structures to the ends of a path (Wang et al., 2021). In random-walk extremal problems, the balanced double broom is singled out by an almost equal split of the pendant leaves between the two ends of a diameter path (Beveridge et al., 2024, Beveridge et al., 4 Aug 2025). These variants support sharp theorems on positivity, Wiener index, exact mixing time, meeting time, reconstruction from edge-deleted data, and depth invariants of edge ideals.

1. Definitions and notational conventions

The literature uses several precise realizations of the same underlying shape: a path-like spine with leaf sets concentrated at two distinguished attachment vertices. This variation in notation suggests that “double broom” functions as a stable structural motif rather than a single universally fixed convention.

Source	Notation	Defining description
Degree-associated edge reconstruction	$D_{m,n,p}$	Tree with $m+n+p$ vertices obtained from a $p$-vertex path by appending $m$ leaf neighbors to one end and $n$ leaf neighbors to the other end (Ma et al., 2016)
Wiener index of trees	$p$0	Tree on $p$1 vertices with exactly two broom vertices $p$2, with $p$3 and $p$4 (Borovićanin et al., 23 Jul 2025)
Chromatic symmetric functions	$p$5	Family built by joining two broom-like end structures to the ends of a path; concretely obtained by identifying the center of $p$6 with the leaf on the long leg of $p$7 (Wang et al., 2021)
Strong double broom reconstruction	$p$8	Graph obtained from internally vertex-disjoint $p$9-paths with common ends $m$0, then appending leaves at $m$1 and $m$2 (Anushadevi et al., 2018)
Edge-ideal literature	$m$3	Double broom graph built from a left path, a middle path, and a right path, with edge ideal $m$4 (Bordianu et al., 2024)

Two specialized variants recur in recent extremal work. The balanced double broom is the double broom in which the two end leaf counts differ by at most one (Beveridge et al., 2024, Borovićanin et al., 23 Jul 2025). The balanced near double broom is a parity-corrected variant in which one additional singleton leaf is attached near the middle of the spine (Beveridge et al., 4 Aug 2025). A further extension is the strong double broom, where the single spine is replaced by at least two internally vertex-disjoint paths with the same ends (Anushadevi et al., 2018).

2. Chromatic symmetric functions and positivity

For the chromatic symmetric function

$m$5

the elementary and Schur expansions are

$m$6

A graph is $m$7-positive if all $m$8 are nonnegative, and Schur positive if all $m$9 are nonnegative (Wang et al., 2021).

For the double broom family

$n$0

the classification is particularly rigid: no graph in $n$1 is $n$2-positive, and exactly ten graphs are Schur positive (Wang et al., 2021). Those ten sporadic cases are

$n$3

$n$4

The non-$n$5-positivity proofs combine Orellana–Scott’s triple-deletion property with explicit extraction of carefully chosen $n$6-coefficients. A representative calculation is for $n$7: $n$8 which immediately excludes $n$9-positivity (Wang et al., 2021). For Schur coefficients, the method uses Wang and Wang’s formula in terms of special rim hook tabloids,

$\mathrm{br}'(l,p,l')$0

and then counts semi-ordered stable partitions of the relevant types.

The same paper isolates a subfamily with a contrasting behavior. It proves that $\mathrm{br}'(l,p,l')$1 is never $\mathrm{br}'(l,p,l')$2-positive, but conjectures that for any integer $\mathrm{br}'(l,p,l')$3,

$\mathrm{br}'(l,p,l')$4

is Schur positive, and reports computational verification up to $\mathrm{br}'(l,p,l')$5 (Wang et al., 2021). This provides a concrete instance in which Schur positivity is strictly more permissive than $\mathrm{br}'(l,p,l')$6-positivity.

3. Extremal Wiener index

The Wiener index is

$\mathrm{br}'(l,p,l')$7

For trees of fixed diameter $\mathrm{br}'(l,p,l')$8 and order $\mathrm{br}'(l,p,l')$9, double brooms emerge as extremal objects in a quantitatively delimited regime (Borovićanin et al., 23 Jul 2025).

A broom vertex in a tree is any vertex adjacent to a leaf. A double broom $D_{m,n,p}$0 is then a tree with exactly two broom vertices $D_{m,n,p}$1 and $D_{m,n,p}$2 such that

$D_{m,n,p}$3

Equivalently, all leaves are attached to one of two vertices, and the remainder of the tree is the path connecting them (Borovićanin et al., 23 Jul 2025).

The central theorem states that if $D_{m,n,p}$4 is a tree of diameter $D_{m,n,p}$5 on

$D_{m,n,p}$6

vertices and

$D_{m,n,p}$7

then $D_{m,n,p}$8 is maximum if and only if $D_{m,n,p}$9 is a double broom graph (Borovićanin et al., 23 Jul 2025). The result is sharp up to a small constant. Specifically, if

$m+n+p$0

then there exists a triple broom $m+n+p$1 of diameter $m+n+p$2 such that for every double broom $m+n+p$3 of diameter $m+n+p$4,

$m+n+p$5

The proof uses a structural reduction centered on special vertices and broom relocation. A leaf-at-diameter-end lemma forces every leaf in a Wiener-maximal tree to have eccentricity $m+n+p$6. A key relocation formula compares $m+n+p$7 after moving one broom component to the opposite side of a special vertex, and leaf-relocation arguments then enforce a near-balance condition

$m+n+p$8

on symmetric parts of an extremal tree (Borovićanin et al., 23 Jul 2025). The theorem’s contradiction mechanism shows that, in the prescribed regime, any maximal tree that is not already a double broom would necessarily contain a forbidden special-vertex configuration.

Among double brooms with fixed total number of pendant leaves $m+n+p$9, the most balanced member maximizes the Wiener index: if $p$0 is even, the maximum occurs at $p$1; if $p$2 is odd, it occurs at

$p$3

(Borovićanin et al., 23 Jul 2025). This balanced-split principle recurs in random-walk extremal problems as well.

4. Random walks: exact mixing time and meeting time

For trees of fixed order and diameter, double brooms occupy opposite extremal roles for two distinct random-walk functionals.

For exact mixing time, the relevant quantity is

$p$4

where $p$5 is the minimum expected length of a $p$6-stopping rule, and Lovász–Winkler’s identity gives

$p$7

when $p$8 is a $p$9-pessimal vertex (Beveridge et al., 2024). In the family $m$0 of trees of order $m$1 and diameter $m$2, the unique maximizer of $m$3 is the balanced double broom. In that paper, a double broom consists of a path $m$4 with $m$5 pendant edges incident with $m$6 and $m$7 pendant edges incident with $m$8, where one of these leaves is labeled $m$9 and one is labeled $n$0, with

$n$1

The balanced double broom has

$n$2

and the extremal value is

$n$3

$n$4

(Beveridge et al., 2024). The proof proceeds by surgeries: first reduce to caterpillars, then move interior leaves outward to obtain a double broom or near double broom, then rebalance the end leaf counts.

For meeting time, the functional is

$n$5

equivalently the maximum of $n$6, with the joining-time normalization

$n$7

(Beveridge et al., 4 Aug 2025). Here the extremal picture is reversed. For fixed order $n$8 and diameter $n$9, the meeting time is maximized by the broom graph, whereas it is minimized by the balanced double broom, or by a balanced near double broom, depending on the parity of $p$00 and $p$01. When $p$02 and $p$03 have opposite parity, the balanced double broom has end leaf counts

$p$04

and is the unique minimizer (Beveridge et al., 4 Aug 2025). When $p$05 and $p$06 have the same parity, the minimizer is the balanced near double broom $p$07, obtained by giving each end

$p$08

leaves and attaching one extra singleton leaf adjacent to $p$09.

The explicit endpoint formulas for a double broom with spine $p$10 are

$p$11

$p$12

These show that smaller imbalance between $p$13 and $p$14 lowers the larger of the two endpoint joining times (Beveridge et al., 4 Aug 2025). A plausible implication is that the balanced double broom is not merely an aesthetically symmetric representative of the family, but the precise structure that optimizes several competing transport quantities under fixed order and diameter constraints.

5. Reconstruction theory

The reconstruction literature studies how many degree-annotated edge deletions are needed to determine a double broom uniquely. In this setting, an edge-card is $p$15, the degree of an edge $p$16 is

$p$17

and a degree-associated edge-card, or decard, is the pair

$p$18

The degree-associated edge-reconstruction number $p$19 is the least number of decards sufficient to reconstruct $p$20, while $p$21 is the least $p$22 such that every set of $p$23 decards determines $p$24 (Ma et al., 2016).

For the ordinary double broom $p$25, the edges are divided into leaf edges, middle edges, and hub edges. The classification theorem states that $p$26 is always $p$27 or $p$28, with $p$29 exactly when there is an edge satisfying the one-decard criterion of Lemma 2.1, and $p$30 otherwise (Ma et al., 2016). The adversary parameter has a full case-by-case classification. Its generic value is

$p$31

but there are exceptional families with values $p$32, $p$33, $p$34, and $p$35. The value $p$36 occurs for $p$37, for $p$38 with $p$39, and for $p$40 with $p$41; the value $p$42 occurs for $p$43, $p$44, and $p$45 with $p$46; many asymmetric and near-symmetric families lie in the $p$47 class (Ma et al., 2016). For the special case $p$48, the double broom becomes a double-star, and the corollary gives

$p$49

while

$p$50

The strong-double-broom generalization replaces the single spine by multiple internally vertex-disjoint $p$51-paths. A strong double broom is a graph on at least $p$52 vertices obtained from a union of at least two internally vertex-disjoint paths with the same ends $p$53 and $p$54, by appending leaves at $p$55 and $p$56 (Anushadevi et al., 2018). Its da-ecards are classified as leaf da-ecards $p$57, middle da-ecards $p$58, and hub da-ecards $p$59. For every strong double broom $p$60,

$p$61

and for the symmetric family $p$62,

$p$63

in most cases, with explicit exceptions where it is $p$64 or $p$65, and one notable family where

$p$66

(Anushadevi et al., 2018). These results show that double-broom structures are unusually rigid under degree-associated edge deletion, even when the spine is replaced by parallel internally disjoint paths.

6. Edge ideals and relation to broom-only literatures

In commutative algebra, the double broom graph $p$67 is defined on

$p$68

with edge ideal

$p$69

where $p$70, $p$71, and $p$72 encode a left path, a middle path, and a right path, respectively (Bordianu et al., 2024). For $p$73, the main depth theorem is

$p$74

For the ideal itself,

$p$75

(Bordianu et al., 2024). The proof uses the colon ideal

$p$76

which reduces $p$77 to a polynomial extension of a shorter path quotient. In this formulation, the double broom serves as a bridge between path-graph depth formulas and more structured tree ideals.

A recurrent source of confusion is that not every broom-related paper actually defines a double broom. In algorithmic graph theory, the object defined is the $p$78-broom $p$79, consisting of a path $p$80 and $p$81 additional leaves attached to one endpoint; the paper does not define a double broom at all, and its broom-related theorem is a $p$82-approximation for Maximum Independent Set on $p$83-vertex $p$84-free graphs in time

$p$85

(Bacsó et al., 2018). Likewise, the extremal paper on forbidden brooms studies only the single broom $p$86, obtained from an $p$87-vertex path by adding $p$88 leaves adjacent to a penultimate vertex, and explicitly does not define or study a double broom (Gerbner, 2024). This suggests that the double broom occupies a distinct place in the literature: it is not merely a routine two-ended analogue of a broom, but a separate object whose extremal, algebraic, and reconstructive behavior has required its own dedicated analyses.