(h,k)-Quasi-Double Star in Planar Extremal Graphs

• The (h,k)-quasi-double star is a caterpillar tree defined by a 3-vertex spine with h leaves attached to one end and k leaves to the other.
• Planar Turán number analysis for W₍ₕ,ₖ₎ employs block constructions and inductive proofs to derive tight edge bounds across different h and k regimes.
• These findings extend extremal graph theory by elucidating how local degree constraints in planar graphs govern the maximum edge count in forbidden subgraph settings.

The $(h,k)$-quasi-double star $W_{h,k}$ is the tree obtained from the path $P_3=v_1v_2v_3$ by attaching $h$ leaves to $v_1$ and $k$ leaves to $v_3$. Equivalently, it is denoted $P_3(h,0,k)$. As recorded by Liu, Xie and Zhao, $W_{h,k}$ is a subclass of caterpillars, specifically a caterpillar whose spine is the 3-vertex path $v_1v_2v_3$; the same work studies the planar Turán number $W_{h,k}$0 for all $W_{h,k}$1 (Liu et al., 16 Jul 2025).

1. Definition and notation

Let $W_{h,k}$2 be the path on three vertices. For nonnegative integers $W_{h,k}$3, attach $W_{h,k}$4 new leaves

$W_{h,k}$5

to $W_{h,k}$6 and $W_{h,k}$7 new leaves

$W_{h,k}$8

to $W_{h,k}$9. The resulting tree is denoted

$P_3=v_1v_2v_3$0

and is called the $P_3=v_1v_2v_3$1-quasi-double star.

Its vertex and edge sets are

$P_3=v_1v_2v_3$2

$P_3=v_1v_2v_3$3

This presentation isolates the two attachment sites $P_3=v_1v_2v_3$4 and $P_3=v_1v_2v_3$5 at the ends of the spine. In the parameter range treated for planar Turán purposes, the paper fixes $P_3=v_1v_2v_3$6, but the combinatorial definition itself is stated for arbitrary nonnegative integers $P_3=v_1v_2v_3$7.

2. Basic structural properties

The order and size of $P_3=v_1v_2v_3$8 are immediate: $P_3=v_1v_2v_3$9 The derivation is direct: one starts with 3 vertices and 2 edges in the path $h$0, and each added leaf contributes one vertex and one edge.

The degree sequence is equally explicit. In $h$1,

$h$2

while

$h$3

When all leaves are removed, the remaining subgraph is exactly the path $h$4. Hence $h$5 is a caterpillar; more precisely, it is a special caterpillar with spine of length $h$6. This identifies the quasi-double star as a highly constrained tree family in which all non-spine branching is concentrated at the two endpoints of a 3-vertex path.

3. Representative small instances

Two small examples from the reference summarize the combinatorial shape effectively.

For $h$7, the graph has vertex set

$h$8

so it has six vertices in total. Its edge set is

$h$9

The distinguished spine remains $v_1$0, with one leaf at $v_1$1 and two leaves at $v_1$2. The corresponding degrees are

$v_1$3

For $v_1$4, the graph has $v_1$5 vertices and $v_1$6 edges. Again the spine is $v_1$7, now with two leaves at $v_1$8 and four at $v_1$9. This example is especially relevant because $k$0 lies on the $k$1 boundary where the planar Turán behavior differs from the exact formula valid for $k$2.

These examples make clear that the family is indexed by the two endpoint leaf-multiplicities, while the internal vertex $k$3 always has degree $k$4.

4. Planar Turán numbers for quasi-double stars

For a graph $k$5, the planar Turán number is

$k$6

that is, the maximum number of edges in an $k$7-free planar graph on $k$8 vertices.

For quasi-double stars, the main theorem of Liu, Xie and Zhao fixes $k$9 and gives the following bounds (Liu et al., 16 Jul 2025).

Parameter regime	Bound for $v_3$0	Tightness statement
$v_3$1	$v_3$2	Equality holds when $v_3$3
$v_3$4	$v_3$5	For $v_3$6, the upper bound is tight whenever $v_3$7
$v_3$8	$v_3$9	Remaining case

In particular, when $P_3(h,0,k)$0,

$P_3(h,0,k)$1

The theorem exhibits three distinct regimes. The first regime, $P_3(h,0,k)$2, admits an exact linear coefficient $P_3(h,0,k)$3 together with a divisibility-based equality condition. The second regime, $P_3(h,0,k)$4, yields a nontrivial interval between $P_3(h,0,k)$5 and $P_3(h,0,k)$6, except that the case $P_3(h,0,k)$7 has a sharp upper bound under the stated divisibility condition. The remaining case $P_3(h,0,k)$8 has a larger admissible density window, namely from $P_3(h,0,k)$9 up to $W_{h,k}$0.

5. Extremal constructions and proof strategy

The lower bounds for $W_{h,k}$1 are obtained via block-copied planar graphs. When $W_{h,k}$2, one partitions the $W_{h,k}$3 vertices into blocks of size $W_{h,k}$4 and makes each block a maximal planar graph. Such a block has

$W_{h,k}$5

edges, and summing over the blocks gives the lower bound

$W_{h,k}$6

The theorem states that equality is achieved in this way.

For $W_{h,k}$7, the sharp construction uses a $W_{h,k}$8-regular triangulation on $W_{h,k}$9 vertices. The paper states that there is a unique planar triangulation on $v_1v_2v_3$0 vertices all of degree $v_1v_2v_3$1, up to isomorphism, and that it has $v_1v_2v_3$2 edges. Taking $v_1v_2v_3$3 disjoint copies yields exactly $v_1v_2v_3$4 edges and is $v_1v_2v_3$5-free, since $v_1v_2v_3$6 would need a vertex of degree $v_1v_2v_3$7 (Liu et al., 16 Jul 2025).

The upper bounds are derived by induction and local structure analysis. The authors develop neighborhood-analysis lemmas, including Lemmas 2.4–2.8, showing that in a planar $v_1v_2v_3$8-free graph, vertices of large degree “trap” entire components in their closed neighborhoods. The argument then peels off such components or low-degree vertices and applies the inductive hypothesis to the remainder, with the edge accounting calibrated to the target linear coefficient.

The proof toolkit also includes the planar inequalities

$v_1v_2v_3$9

and

$W_{h,k}$00

together with a glue-and-delete argument identified as Lemma 2.7. According to the summary, this lemma shows that deleting a small planar component never raises the average edge density above a target $W_{h,k}$01. The overall methodology is therefore an overview of local obstruction analysis, recursive decomposition, and standard planar extremal estimates.

6. Role within planar extremal graph theory

The quasi-double star results are positioned as an extension of the program of determining planar Turán numbers for small trees, including double-stars, short cycles, and theta-graphs (Liu et al., 16 Jul 2025). One cited antecedent in this line is “Planar Turán number of double stars” (Ghosh et al., 2021).

Within that program, the family $W_{h,k}$02 is significant because it interpolates between very small caterpillar obstructions and denser forbidden-tree configurations. The summary emphasizes that the ratio

$W_{h,k}$03

jumps from

$W_{h,k}$04

up to $W_{h,k}$05 or beyond as the “size” $W_{h,k}$06 grows. The tight constructions also underscore the special role of regular triangulations in forbidding certain small subtrees while maximizing edge count.

A plausible implication is that quasi-double stars provide a useful test family for understanding how local degree constraints interact with planar density. Because $W_{h,k}$07 is simultaneously a tree, a caterpillar, and a configuration with two asymmetric branching sites, it isolates the extent to which planar extremal behavior is controlled by endpoint degrees rather than by longer spine structure.