Balanced Double Broom Graph

• The balanced double broom graph is defined as a symmetric tree where pendant edges are equally distributed at two ends, forming dual broom heads.
• It encompasses multiple formulations such as balanced double stars, path-end double brooms, and strong double brooms with source-dependent definitions.
• It plays a key role in planar Turán theory, random walk extremal studies, and graph reconstruction, yielding precise combinatorial and extremal results.

The balanced double broom graph is a symmetric broom-like tree construction whose precise formalization is source-dependent. In recent arXiv literature, the term is used either explicitly or by close equivalence for several related families: the balanced double star $S_{k,k}$, two-ended diameter-constrained trees denoted $D_{n,d}$, path-end double-brooms $D_{m,m,p}$, and the more general strong double broom $B(n,n,mP_k)$ with multiple internally vertex-disjoint hub-to-hub paths (Xu et al., 2024, Beveridge et al., 28 Oct 2025, Ma et al., 2016, Anushadevi et al., 2018). Across these formulations, the common geometric idea is the same: pendant structure is distributed symmetrically, or as symmetrically as possible, at the two ends of a path or between two hubs.

1. Terminology and source-specific definitions

The literature represented here does not impose a single universal definition of “balanced double broom graph.” Instead, the name is attached to several closely related symmetric constructions.

Source	Notation	Defining construction
(Xu et al., 2024)	$S_{m,m}$	Take an edge $xy$, join $x$ with $m$ vertices and $y$ with $m$ distinct vertices; this is the balanced double star
(Beveridge et al., 28 Oct 2025)	$D_{n,d}$0	A path $D_{n,d}$1 with $D_{n,d}$2 pendant edges at $D_{n,d}$3 and $D_{n,d}$4 at $D_{n,d}$5, balanced by $D_{n,d}$6, $D_{n,d}$7
(Beveridge et al., 4 Aug 2025)	$D_{n,d}$8	A path $D_{n,d}$9 with $D_{m,m,p}$0 leaves at $D_{m,m,p}$1 and $D_{m,m,p}$2 leaves at $D_{m,m,p}$3, balanced by $D_{m,m,p}$4, $D_{m,m,p}$5
(Ma et al., 2016)	$D_{m,m,p}$6	A $D_{m,m,p}$7-vertex path with $D_{m,m,p}$8 leaves appended at each end
(Anushadevi et al., 2018)	$D_{m,m,p}$9	Two hubs $B(n,n,mP_k)$0, $B(n,n,mP_k)$1 leaves at each hub, and $B(n,n,mP_k)$2 internally vertex-disjoint $B(n,n,mP_k)$3-$B(n,n,mP_k)$4 paths of order $B(n,n,mP_k)$5

The balanced double star $B(n,n,mP_k)$6 is the simplest of these models. The paper defining double stars states that a double star $B(n,n,mP_k)$7 is obtained by taking an edge $B(n,n,mP_k)$8 and joining $B(n,n,mP_k)$9 with $S_{m,m}$0 vertices and $S_{m,m}$1 with $S_{m,m}$2 distinct vertices, and that the graph is balanced when $S_{m,m}$3; consequently $S_{m,m}$4 and $S_{m,m}$5 (Xu et al., 2024).

The path-end double-broom model $S_{m,m}$6 is equally explicit: it is the tree with $S_{m,m}$7 vertices obtained from a $S_{m,m}$8-vertex path by appending $S_{m,m}$9 leaf neighbors at one end and $xy$0 at the other (Ma et al., 2016). The strong balanced double broom $xy$1 broadens this by allowing a bundle of $xy$2 internally vertex-disjoint $xy$3-$xy$4 paths of equal order $xy$5 rather than a single central path (Anushadevi et al., 2018).

A notable negative fact is also part of the terminology. The $xy$6-boundedness paper on $xy$7-brooms does not define or mention a balanced double broom graph; its subject is the one-sided $xy$8-broom, not a two-sided balanced variant (Liu et al., 2021). This suggests a source-dependent convention rather than a universally fixed object.

2. Relation to brooms, double stars, and multibrooms

The closest one-sided comparator is the $xy$9-broom. For a positive integer $x$0, a $x$1-broom is the graph obtained from $x$2 by subdividing an edge once. With the notation of that paper, its vertex set is

$x$3

and its edge set is

$x$4

It is therefore a subdivided star with one high-degree center, one degree-$x$5 vertex on a path of length $x$6, and $x$7 additional pendant leaves. The same source records the small cases $x$8-broom $x$9 and $m$0-broom $m$1 chair or fork (Liu et al., 2021). A balanced double broom is not this object: the $m$2-broom is intrinsically one-sided, whereas the double-broom constructions above are two-sided.

The broom language used in induced-subgraph theory is broader. A broom of length $m$3 is obtained from a $m$4-edge path with ends $m$5 by adding leaves adjacent to $m$6, and $m$7 is called the handle. A $m$8-broom is such a broom with exactly $m$9 leaves, and a $y$0-multibroom is obtained by identifying the handles of $y$1 such brooms (Scott et al., 2018). In that framework, a balanced double broom is most naturally modeled as a $y$2-multibroom, and if equal leaf multiplicities are also intended, by identifying the handles of two isomorphic $y$3-brooms. That identification is an interpretation rather than terminology fixed by the paper itself (Scott et al., 2018).

A second nearby one-sided model is the broom $y$4, defined as the graph obtained from an $y$5-vertex path by adding $y$6 new leaves connected to a penultimate vertex $y$7 of the path, where $y$8 is called the center of the broom (Gerbner, 2024). In that sense, the balanced double broom is the two-sided analogue of a one-sided path-with-leaves tree. This analogy is methodological rather than terminological: the paper on $y$9 does not state a theorem for balanced double brooms, but its extremal constructions and common-neighborhood arguments are explicitly presented as highly relevant to such two-sided variants (Gerbner, 2024).

A common misconception is therefore easy to isolate. The terms $m$0-broom, broom $m$1, double star $m$2, multibroom $m$3, and balanced double broom do not coincide across these papers. They form a family of related but nonidentical broom-shaped trees, differing chiefly in whether branching occurs at one side, at two adjacent centers, or at the ends of a longer path (Liu et al., 2021, Scott et al., 2018, Gerbner, 2024).

3. Balanced double stars and planar Turán theory

In planar extremal graph theory, the relevant balanced object is the balanced double star $m$4. The planar Turán number $m$5 is defined as the maximum number of edges in an $m$6-vertex planar graph with no $m$7 subgraph, and the balanced double star paper gives an exact classification for this family (Xu et al., 2024).

The paper emphasizes that, among balanced double stars, $m$8 was the only remaining unresolved case. It then proves the exact value of $m$9, thereby completing the planar Turán theory for all balanced double stars (Xu et al., 2024).

Balanced double star	Exact planar Turán number	Regime
$D_{n,d}$00	$D_{n,d}$01 if $D_{n,d}$02, else $D_{n,d}$03	all $D_{n,d}$04
$D_{n,d}$05	$D_{n,d}$06	$D_{n,d}$07
$D_{n,d}$08	$D_{n,d}$09, $D_{n,d}$10, $D_{n,d}$11, or $D_{n,d}$12	piecewise in $D_{n,d}$13
$D_{n,d}$14 with $D_{n,d}$15	$D_{n,d}$16	all $D_{n,d}$17

For $D_{n,d}$18, the exact formula is

$D_{n,d}$19

The same paper records that $D_{n,d}$20 has two adjacent center vertices, each with three pendant neighbors, so it has $D_{n,d}$21 vertices and $D_{n,d}$22 edges, and that $D_{n,d}$23 is a path on $D_{n,d}$24 vertices (Xu et al., 2024).

The extremal significance of the balanced double star is therefore exact rather than asymptotic in the planar setting. For $D_{n,d}$25, the problem becomes trivial in the planar class because a double wheel avoids $D_{n,d}$26, giving $D_{n,d}$27; for $D_{n,d}$28, the exact value is genuinely nontrivial and piecewise (Xu et al., 2024). This sharply distinguishes the balanced double star from many other broom-like trees whose planar extremal behavior remains only partially understood.

4. Random walks, meeting times, and diameter-constrained balanced double brooms

Two recent random-walk papers assign opposite extremal roles to balanced double broom trees because they optimize different functionals.

Objective	Extremal role of the balanced double broom	Source
$D_{n,d}$29	unique maximizer in $D_{n,d}$30	(Beveridge et al., 28 Oct 2025)
$D_{n,d}$31	unique minimizer in $D_{n,d}$32 when $D_{n,d}$33 have opposite parity; otherwise replaced by a balanced near double broom	(Beveridge et al., 4 Aug 2025)

For a tree $D_{n,d}$34, both papers use the stationary distribution

$D_{n,d}$35

and the hitting time $D_{n,d}$36, the expected number of steps needed for a random walk started at $D_{n,d}$37 to reach $D_{n,d}$38 (Beveridge et al., 28 Oct 2025, Beveridge et al., 4 Aug 2025).

In the best-meeting-time paper, the balanced double broom $D_{n,d}$39 is the unique maximizer of

$D_{n,d}$40

among trees of order $D_{n,d}$41 and diameter $D_{n,d}$42. The paper defines a double broom in $D_{n,d}$43 as a path $D_{n,d}$44 with $D_{n,d}$45 pendant edges at $D_{n,d}$46 and $D_{n,d}$47 at $D_{n,d}$48, one leaf at each side labeled $D_{n,d}$49 and $D_{n,d}$50, and calls it balanced when

$D_{n,d}$51

Its main theorem states that for $D_{n,d}$52, the maximum best meeting time over $D_{n,d}$53 is achieved uniquely by $D_{n,d}$54, with an explicit parity-dependent closed form (Beveridge et al., 28 Oct 2025).

That paper also identifies the minimizing meeting vertex via barycenters. The minimizers of the joining time are exactly the barycenter(s), so the best meeting vertex in a balanced double broom is a barycenter. It further shows that when the graph is split at a barycenter, the two resulting rooted subtrees are brooms, and the minimum joining time decomposes as a sum of the corresponding maximum joining times of those brooms (Beveridge et al., 28 Oct 2025).

The meeting-time paper studies the worst-target functional

$D_{n,d}$55

Here the balanced double broom is the minimizing shape for fixed $D_{n,d}$56 and $D_{n,d}$57 only when $D_{n,d}$58 and $D_{n,d}$59 have opposite parity. In that case,

$D_{n,d}$60

When $D_{n,d}$61 and $D_{n,d}$62 have the same parity, the minimizer becomes a balanced near double broom $D_{n,d}$63, obtained by moving the unavoidable extra leaf from an endpoint to the middle of the spine (Beveridge et al., 4 Aug 2025).

The contrast with the broom graph is exact. In the meeting-time paper, the broom $D_{n,d}$64 uniquely maximizes $D_{n,d}$65, whereas the balanced double broom or balanced near double broom minimizes it (Beveridge et al., 4 Aug 2025). In the best-meeting-time paper, the balanced double broom uniquely maximizes $D_{n,d}$66 for fixed $D_{n,d}$67, and among all trees on $D_{n,d}$68 vertices the path $D_{n,d}$69 is the maximizer when $D_{n,d}$70 is even, while $D_{n,d}$71 is the maximizer when $D_{n,d}$72 is odd and $D_{n,d}$73 (Beveridge et al., 28 Oct 2025). This objective-function dependence is central: the same balanced tree shape can be extremal in opposite directions for different random-walk criteria.

5. Reconstruction theory for double-broom families

The reconstruction literature fixes a more rigid path-end model. The double-broom $D_{n,d}$74 is the tree obtained from a $D_{n,d}$75-vertex path by appending $D_{n,d}$76 leaf neighbors at one end and $D_{n,d}$77 at the other, and the balanced double broom is the symmetric case $D_{n,d}$78 (Ma et al., 2016).

The two reconstruction parameters are the degree-associated edge-reconstruction number $D_{n,d}$79, the minimum number of decards sufficient to reconstruct $D_{n,d}$80, and the adversary degree-associated edge-reconstruction number $D_{n,d}$81, the least $D_{n,d}$82 such that every set of $D_{n,d}$83 decards determines $D_{n,d}$84 (Ma et al., 2016). For balanced double-brooms, the paper provides a complete classification of $D_{n,d}$85 and a criterion-based classification of $D_{n,d}$86.

Balanced double-broom	$D_{n,d}$87
$D_{n,d}$88	$D_{n,d}$89
$D_{n,d}$90, $D_{n,d}$91	$D_{n,d}$92
$D_{n,d}$93	$D_{n,d}$94
$D_{n,d}$95, $D_{n,d}$96	$D_{n,d}$97
$D_{n,d}$98, $D_{n,d}$99	$D_{m,m,p}$00
$D_{m,m,p}$01, $D_{m,m,p}$02	$D_{m,m,p}$03
$D_{m,m,p}$04, $D_{m,m,p}$05	$D_{m,m,p}$06
$D_{m,m,p}$07, $D_{m,m,p}$08	$D_{m,m,p}$09
$D_{m,m,p}$10, $D_{m,m,p}$11	$D_{m,m,p}$12
$D_{m,m,p}$13, $D_{m,m,p}$14	$D_{m,m,p}$15
$D_{m,m,p}$16, $D_{m,m,p}$17	$D_{m,m,p}$18

For $D_{m,m,p}$19, the same paper proves that $D_{m,m,p}$20 is always $D_{m,m,p}$21 or $D_{m,m,p}$22, with

$D_{m,m,p}$23

Specializing the balanced cases extracted in the source, one has

$D_{m,m,p}$24

and for all $D_{m,m,p}$25,

$D_{m,m,p}$26

All remaining balanced cases have $D_{m,m,p}$27 (Ma et al., 2016).

The strong double broom paper generalizes the symmetric model to

$D_{m,m,p}$28

with two hubs $D_{m,m,p}$29, $D_{m,m,p}$30 leaves at each hub, and $D_{m,m,p}$31 internally vertex-disjoint $D_{m,m,p}$32-$D_{m,m,p}$33 paths of order $D_{m,m,p}$34. Its basic parameters are

$D_{m,m,p}$35

with two hub vertices of degree $D_{m,m,p}$36, $D_{m,m,p}$37 leaves of degree $D_{m,m,p}$38, and $D_{m,m,p}$39 internal path vertices of degree $D_{m,m,p}$40 (Anushadevi et al., 2018).

For this balanced strong double broom, the paper proves that $D_{m,m,p}$41 is always $D_{m,m,p}$42 or $D_{m,m,p}$43, and determines $D_{m,m,p}$44. The exceptional value is

$D_{m,m,p}$45

while $D_{m,m,p}$46 is $D_{m,m,p}$47 in most remaining cases and $D_{m,m,p}$48 or $D_{m,m,p}$49 in the rest (Anushadevi et al., 2018). The symmetry of the balanced construction is structurally important here because leaf, hub, and middle da-ecards come in repeated isomorphism classes, making adversarial reconstruction subtler than existential reconstruction.

6. Balancedness, caterpillar structure, and source-dependent limits

A different use of “balanced” arises in clique-matrix theory. A graph is balanced if its clique-matrix contains no square submatrix of odd order with exactly two $D_{m,m,p}$50's in each row and column. Within the class of distance-hereditary graphs, balanced graphs are exactly the hereditary clique-Helly graphs, equivalently the graphs with no induced $D_{m,m,p}$51 (Busolini et al., 1 Jul 2026). Standard double broom graphs are trees, hence distance-hereditary, and therefore balanced in this sense because a tree cannot contain the dense six-vertex graph $D_{m,m,p}$52 (Busolini et al., 1 Jul 2026). This is a different notion from “balanced double broom,” but it gives an exact structural characterization for the usual tree forms.

Balanced domination supplies another specialized framework. A balanced domination function is a labeling $D_{m,m,p}$53 such that the sum of labels over every closed neighborhood is zero, and the balanced domination number is

$D_{m,m,p}$54

A graph is called $D_{m,m,p}$55-balanced when $D_{m,m,p}$56 (Nikolic et al., 9 Nov 2025). That paper does not provide a closed formula for balanced double broom graphs, but it does treat caterpillars, and a double broom is a special caterpillar. For a caterpillar $D_{m,m,p}$57, any nonzero modified balanced domination function forces the necessary congruence

$D_{m,m,p}$58

where $D_{m,m,p}$59 is the total number of leaves (Nikolic et al., 9 Nov 2025). Specializing to a double broom with spine $D_{m,m,p}$60 and endpoint leaf counts $D_{m,m,p}$61, this gives the necessary condition $D_{m,m,p}$62. The same source records the tridiagonal spine system

$D_{m,m,p}$63

as an immediate specialization of the caterpillar equations (Nikolic et al., 9 Nov 2025).

Finally, several papers in the supplied corpus are methodologically adjacent without directly defining the object. The $D_{m,m,p}$64-broom $D_{m,m,p}$65-boundedness paper treats a one-sided subdivided star, not a balanced double broom (Liu et al., 2021). The broom extremal paper treats the one-sided broom $D_{m,m,p}$66, not a two-sided balanced form, but explicitly frames its results as a structural precursor for balanced double broom problems (Gerbner, 2024). The multibroom paper covers $D_{m,m,p}$67-multibrooms and therefore includes some balanced double broom instances of types $D_{m,m,p}$68 and $D_{m,m,p}$69, but not all possible balanced side lengths (Scott et al., 2018).

Taken together, these sources show that “balanced double broom graph” is best understood as a family resemblance rather than a single invariant definition. In planar Turán theory it is effectively the balanced double star $D_{m,m,p}$70; in fixed-diameter random-walk extremal theory it is the two-ended tree $D_{m,m,p}$71; in reconstruction theory it is the symmetric path-end tree $D_{m,m,p}$72; and in strong reconstruction it is the hub-symmetric graph $D_{m,m,p}$73. The unifying feature is bilateral symmetry of the broom heads, while the exact combinatorial model is determined by the ambient problem (Xu et al., 2024, Beveridge et al., 28 Oct 2025, Ma et al., 2016, Anushadevi et al., 2018).