---
title: Balanced Double Broom Graph
url: https://www.emergentmind.com/topics/balanced-double-broom-graph
type: topic
---

# Balanced Double Broom Graph

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 [2406.05758] [2510.24387] [1604.02908] [1803.01582]. 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 |
|---|---|---|
| [2406.05758] | \(S_{m,m}\) | Take an edge \(xy\), join \(x\) with \(m\) vertices and \(y\) with \(m\) distinct vertices; this is the balanced double star |
| [2510.24387] | \(D_{n,d}\) | A path \(v_1,\ldots,v_{d-1}\) with \(\ell\) pendant edges at \(v_1\) and \(r\) at \(v_{d-1}\), balanced by \(\ell=\lfloor (n-d-1)/2\rfloor\), \(r=\lceil (n-d-1)/2\rceil\) |
| [2508.02804] | \(D_{n,d}\) | A path \(v_1,\ldots,v_{d-1}\) with \(\ell\) leaves at \(v_1\) and \(r\) leaves at \(v_{d-1}\), balanced by \(\ell=\lfloor (n-d+1)/2\rfloor\), \(r=\lceil (n-d+1)/2\rceil\) |
| [1604.02908] | \(D_{m,m,p}\) | A \(p\)-vertex path with \(m\) leaves appended at each end |
| [1803.01582] | \(B(n,n,mP_k)\) | Two hubs \(u,v\), \(n\) leaves at each hub, and \(m\) internally vertex-disjoint \(u\)-\(v\) paths of order \(k\) |

The balanced double star \(S_{m,m}\) is the simplest of these models. The paper defining double stars states that a double star \(S_{m,l}\) is obtained by taking an edge \(xy\) and joining \(x\) with \(m\) vertices and \(y\) with \(l\) distinct vertices, and that the graph is balanced when \(m=l\); consequently \(v(S_{m,l})=m+l+2\) and \(e(S_{m,l})=m+l+1\) [2406.05758].

The path-end double-broom model \(D_{m,m,p}\) is equally explicit: it is the tree with \(m+m+p\) vertices obtained from a \(p\)-vertex path by appending \(m\) leaf neighbors at one end and \(m\) at the other [1604.02908]. The strong balanced double broom \(B(n,n,mP_k)\) broadens this by allowing a bundle of \(m\) internally vertex-disjoint \(u\)-\(v\) paths of equal order \(k\) rather than a single central path [1803.01582].

A notable negative fact is also part of the terminology. The \(\chi\)-boundedness paper on \(t\)-brooms does **not** define or mention a balanced double broom graph; its subject is the one-sided \(t\)-broom, not a two-sided balanced variant [2106.08871]. 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 \(t\)-broom. For a positive integer \(t\), a \(t\)-broom is the graph obtained from \(K_{1,t+1}\) by subdividing an edge once. With the notation of that paper, its vertex set is
\[
\{u_0,v_1,v_2,u_1,\dots,u_t\},
\]
and its edge set is
\[
\{u_0v_1,\ v_1v_2\}\cup \{u_0u_i: i\in[t]\}.
\]
It is therefore a subdivided star with one high-degree center, one degree-\(2\) vertex on a path of length \(2\), and \(t\) additional pendant leaves. The same source records the small cases \(1\)-broom \(=P_4\) and \(2\)-broom \(=\) chair or fork [2106.08871]. A balanced double broom is not this object: the \(t\)-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 \(k\) is obtained from a \(k\)-edge path with ends \(a,b\) by adding leaves adjacent to \(b\), and \(a\) is called the handle. A \((k,\delta)\)-broom is such a broom with exactly \(\delta\) leaves, and a \((k_1,\ldots,k_n)\)-multibroom is obtained by identifying the handles of \(n\) such brooms [1807.03768]. In that framework, a balanced double broom is most naturally modeled as a \((k,k)\)-multibroom, and if equal leaf multiplicities are also intended, by identifying the handles of two isomorphic \((k,\delta)\)-brooms. That identification is an interpretation rather than terminology fixed by the paper itself [1807.03768].

A second nearby one-sided model is the broom \(B(\ell,s)\), defined as the graph obtained from an \(\ell\)-vertex path by adding \(s\) new leaves connected to a penultimate vertex \(v\) of the path, where \(v\) is called the center of the broom [2401.11587]. 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 \(B(\ell,s)\) 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 [2401.11587].

A common misconception is therefore easy to isolate. The terms \(t\)-broom, broom \(B(\ell,s)\), double star \(S_{m,m}\), multibroom \((k,k)\), 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 [2106.08871] [1807.03768] [2401.11587].

## 3. Balanced double stars and planar Turán theory

In planar extremal graph theory, the relevant balanced object is the balanced double star \(S_{m,m}\). The planar Turán number \(ex_{\mathcal P}(n,H)\) is defined as the maximum number of edges in an \(n\)-vertex planar graph with no \(H\) subgraph, and the balanced double star paper gives an exact classification for this family [2406.05758].

The paper emphasizes that, among balanced double stars, \(S_{3,3}\) was the only remaining unresolved case. It then proves the exact value of \(ex_{\mathcal P}(n,S_{3,3})\), thereby completing the planar Turán theory for all balanced double stars [2406.05758].

| Balanced double star | Exact planar Turán number | Regime |
|---|---|---|
| \(S_{1,1}\) | \(n\) if \(3\mid n\), else \(n-1\) | all \(n\ge 3\) |
| \(S_{2,2}\) | \(2n-4\) | \(n\ge 16\) |
| \(S_{3,3}\) | \(3n-6\), \(16\), \(18\), or \(\lfloor 5n/2\rfloor-5\) | piecewise in \(n\) |
| \(S_{m,m}\) with \(m\ge 4\) | \(3n-6\) | all \(n\ge 3\) |

For \(S_{3,3}\), the exact formula is
\[
ex_{\mathcal P}(n,S_{3,3})=
\begin{cases}
3n-6,& 3\le n\le 7,\\
16,& n=8,\\
18,& n=9,\\
\left\lfloor \frac{5n}{2}\right\rfloor-5,& n\ge 10.
\end{cases}
\]
The same paper records that \(S_{3,3}\) has two adjacent center vertices, each with three pendant neighbors, so it has \(8\) vertices and \(7\) edges, and that \(S_{1,1}\) is a path on \(4\) vertices [2406.05758].

The extremal significance of the balanced double star is therefore exact rather than asymptotic in the planar setting. For \(m\ge 4\), the problem becomes trivial in the planar class because a double wheel avoids \(S_{m,m}\), giving \(3n-6\); for \(m=3\), the exact value is genuinely nontrivial and piecewise [2406.05758]. 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 |
|---|---|---|
| \(T_{\mathrm{bestmeet}}(G)=\min_{w\in V}\sum_{v\in V}\pi_v H(v,w)\) | unique maximizer in \(\mathcal T_{n,d}\) | [2510.24387] |
| \(T_{\mathrm{meet}}(G)=\max_{w\in V}\sum_{v\in V}\pi_v H(v,w)\) | unique minimizer in \(\mathcal T_{n,d}\) when \(n,d\) have opposite parity; otherwise replaced by a balanced near double broom | [2508.02804] |

For a tree \(G=(V,E)\), both papers use the stationary distribution
\[
\pi_v=\frac{\deg(v)}{2|E|},
\]
and the hitting time \(H(v,w)\), the expected number of steps needed for a random walk started at \(v\) to reach \(w\) [2510.24387] [2508.02804].

In the best-meeting-time paper, the balanced double broom \(D_{n,d}\) is the unique maximizer of
\[
T_{\mathrm{bestmeet}}(G)=\min_{w\in V}\sum_{v\in V}\pi_v H(v,w)
\]
among trees of order \(n\) and diameter \(d\). The paper defines a double broom in \(\mathcal T_{n,d}\) as a path \(v_1,\ldots,v_{d-1}\) with \(\ell\ge 1\) pendant edges at \(v_1\) and \(r\ge 1\) at \(v_{d-1}\), one leaf at each side labeled \(v_0\) and \(v_d\), and calls it balanced when
\[
\ell=\left\lfloor \frac{n-d-1}{2}\right\rfloor,\qquad
r=\left\lceil \frac{n-d-1}{2}\right\rceil.
\]
Its main theorem states that for \(2\le d\le n-1\), the maximum best meeting time over \(\mathcal T_{n,d}\) is achieved uniquely by \(D_{n,d}\), with an explicit parity-dependent closed form [2510.24387].

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 [2510.24387].

The meeting-time paper studies the worst-target functional
\[
T_{\mathrm{meet}}(G)=\max_{w\in V}\sum_{v\in V}\pi_v H(v,w).
\]
Here the balanced double broom is the minimizing shape for fixed \(n\) and \(d\) only when \(n\) and \(d\) have opposite parity. In that case,
\[
\mathcal M(D_{n,d})
=
\frac{1}{2}(d+2)n
+
\frac{d^3-6d^2+8d}{6(n-1)}
-\frac{1}{2}(d+5).
\]
When \(n\) and \(d\) have the same parity, the minimizer becomes a balanced near double broom \(D'_{n,d}\), obtained by moving the unavoidable extra leaf from an endpoint to the middle of the spine [2508.02804].

The contrast with the broom graph is exact. In the meeting-time paper, the broom \(B_{n,d}\) uniquely maximizes \(T_{\mathrm{meet}}\), whereas the balanced double broom or balanced near double broom minimizes it [2508.02804]. In the best-meeting-time paper, the balanced double broom uniquely maximizes \(T_{\mathrm{bestmeet}}\) for fixed \(n,d\), and among all trees on \(n\) vertices the path \(P_n\) is the maximizer when \(n\) is even, while \(B_{n,n-2}=D_{n,n-2}\) is the maximizer when \(n\) is odd and \(n\ge 9\) [2510.24387]. 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 \(D_{m,n,p}\) is the tree obtained from a \(p\)-vertex path by appending \(m\) leaf neighbors at one end and \(n\) at the other, and the balanced double broom is the symmetric case \(D_{m,m,p}\) [1604.02908].

The two reconstruction parameters are the degree-associated edge-reconstruction number \(\dern(G)\), the minimum number of decards sufficient to reconstruct \(G\), and the adversary degree-associated edge-reconstruction number \(\adern(G)\), the least \(k\) such that every set of \(k\) decards determines \(G\) [1604.02908]. For balanced double-brooms, the paper provides a complete classification of \(\adern\) and a criterion-based classification of \(\dern\).

| Balanced double-broom | \(\adern\) |
|---|---:|
| \(D_{1,1,2}\) | \(2\) |
| \(D_{1,1,p}\), \(p\ge 3\) | \(3\) |
| \(D_{2,2,2}\) | \(2\) |
| \(D_{2,2,3}\), \(D_{2,2,4}\) | \(3\) |
| \(D_{2,2,p}\), \(p\ge 5\) | \(4\) |
| \(D_{3,3,2}\), \(D_{3,3,3}\) | \(2\) |
| \(D_{3,3,p}\), \(p\ge 4\) | \(3\) |
| \(D_{m,m,2}\), \(m\ge 4\) | \(1\) |
| \(D_{m,m,3}\), \(m\ge 4\) | \(1\) |
| \(D_{m,m,4}\), \(m\ge 4\) | \(2\) |
| \(D_{m,m,p}\), \(m\ge 4,\ p\ge 5\) | \(3\) |

For \(\dern\), the same paper proves that \(\dern(D_{m,n,p})\) is always \(1\) or \(2\), with
\[
\dern(D_{m,n,p})=
\begin{cases}
1,& \text{if there is an edge satisfying the condition of Lemma 2.1},\\
2,& \text{otherwise}.
\end{cases}
\]
Specializing the balanced cases extracted in the source, one has
\[
\dern(D_{m,m,2})=\dern(D_{m,m,3})=1,
\]
and for all \(m\ge 5\),
\[
\dern(D_{m,m,p})=1\qquad\text{for every }p\ge 2.
\]
All remaining balanced cases have \(\dern=2\) [1604.02908].

The strong double broom paper generalizes the symmetric model to
\[
B(n,n,mP_k),
\]
with two hubs \(u,v\), \(n\) leaves at each hub, and \(m\) internally vertex-disjoint \(u\)-\(v\) paths of order \(k\). Its basic parameters are
\[
|V|=2+2n+m(k-2),\qquad |E|=2n+m(k-1),
\]
with two hub vertices of degree \(n+m\), \(2n\) leaves of degree \(1\), and \(m(k-2)\) internal path vertices of degree \(2\) [1803.01582].

For this balanced strong double broom, the paper proves that \(\dern\) is always \(1\) or \(2\), and determines \(\adern(B(n,n,mP_k))\). The exceptional value is
\[
\adern(B(1,1,2P_k))=5\qquad (k>3),
\]
while \(\adern\) is \(3\) in most remaining cases and \(1\) or \(2\) in the rest [1803.01582]. 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 \(1\)'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 \(\overline{3K_2}\) [2607.00730]. 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 \(\overline{3K_2}\) [2607.00730]. 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 \(f:V(G)\to\{-1,0,1\}\) such that the sum of labels over every closed neighborhood is zero, and the balanced domination number is
\[
\gamma_{bd}(G)=\max\{\omega_f:f\text{ is a BDF on }G\},
\qquad
\omega_f=\sum_{v\in V(G)}f(v).
\]
A graph is called \(d\)-balanced when \(\gamma_{bd}(G)=0\) [2511.06539]. 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 \(C_n\), any nonzero modified balanced domination function forces the necessary congruence
\[
L(C_n)\equiv 3n-2\pmod 4,
\]
where \(L(C_n)\) is the total number of leaves [2511.06539]. Specializing to a double broom with spine \(a_1,\dots,a_n\) and endpoint leaf counts \(s,t\), this gives the necessary condition \(s+t\equiv 3n-2\pmod 4\). The same source records the tridiagonal spine system
\[
(1-s)x_1+x_2=0,\qquad
x_{i-1}+x_i+x_{i+1}=0\ (2\le i\le n-1),\qquad
x_{n-1}+(1-t)x_n=0,
\]
as an immediate specialization of the caterpillar equations [2511.06539].

Finally, several papers in the supplied corpus are methodologically adjacent without directly defining the object. The \(t\)-broom \(\chi\)-boundedness paper treats a one-sided subdivided star, not a balanced double broom [2106.08871]. The broom extremal paper treats the one-sided broom \(B(\ell,s)\), not a two-sided balanced form, but explicitly frames its results as a structural precursor for balanced double broom problems [2401.11587]. The multibroom paper covers \((1,\ldots,1,2,\ldots,2)\)-multibrooms and therefore includes some balanced double broom instances of types \((1,1)\) and \((2,2)\), but not all possible balanced side lengths [1807.03768].

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 \(S_{k,k}\); in fixed-diameter random-walk extremal theory it is the two-ended tree \(D_{n,d}\); in reconstruction theory it is the symmetric path-end tree \(D_{m,m,p}\); and in strong reconstruction it is the hub-symmetric graph \(B(n,n,mP_k)\). The unifying feature is bilateral symmetry of the broom heads, while the exact combinatorial model is determined by the ambient problem [2406.05758] [2510.24387] [1604.02908] [1803.01582].

Source: https://www.emergentmind.com/topics/balanced-double-broom-graph