---
title: 1-Disk OX Drawing in Bipartite 1-Planar Graphs
url: https://www.emergentmind.com/topics/1-disk-ox-drawing
type: topic
---

# 1-Disk OX Drawing in Bipartite 1-Planar Graphs

1-Disk OX Drawing, more precisely a \(1\)-disk \(\mathcal O_X\) drawing, is a constrained \(1\)-planar drawing of a bipartite graph \(G\) with partite sets \(X\) and \(Y\) in which all vertices of \(X\) lie on the boundary of a disk \(\mathcal O\), while all vertices of \(Y\) and all edges lie in the interior of \(\mathcal O\), up to a homeomorphism of the plane. The notion was first proposed by Huang, Ouyang, and Dong in connection with the edge density of bipartite \(1\)-planar graphs, and its current extremal theory is centered on the sharp bound \(|E(G)|\le 2|V(G)|+|X|-6\) for \(2\le |X|\le |Y|\) [2507.19762].

## 1. Definition and topological model

A graph is \(1\)-planar if it admits a drawing in the plane such that each edge is crossed at most once. A graph is bipartite if its vertex set can be partitioned into two subsets \(X\) and \(Y\) such that every edge joins a vertex in \(X\) and a vertex in \(Y\). In a \(1\)-disk \(\mathcal O_X\) drawing, these two conditions are combined with a topological disk constraint: all vertices of \(X\) lie on the boundary \(\partial\mathcal O\), all vertices of \(Y\) lie strictly in the interior of \(\mathcal O\), and all edges are drawn as arcs entirely inside \(\mathcal O\) [2507.19762].

This places the class between ordinary bipartite \(1\)-planar graphs and more specialized outer-face models. Conceptually, it is a kind of outer-bipartite \(1\)-planar drawing: one partite set is forced onto the boundary, while the other partite set and all edge geometry remain internal. The phrase “up to a homeomorphism of the plane” means that the exact Euclidean shape of the disk is irrelevant; only the topological role of a single boundary component matters [2507.19762].

The definition is asymmetric in \(X\) and \(Y\). The partite set \(X\) is the designated boundary set, and the notation \(\mathcal O_X\) records that choice. The proofs exploit this asymmetry heavily, especially the fact that the \(X\)-vertices occur along the boundary of a face and can therefore be duplicated and glued in a controlled way on the sphere [2507.19762].

## 2. Sharp extremal bound

The central theorem is exact. If \(G\) is a bipartite graph with partite sets \(X\) and \(Y\), where \(2\le |X|\le |Y|\), and \(G\) has a \(1\)-disk \(\mathcal O_X\) drawing, then
\[
|E(G)|\le 2|V(G)|+|X|-6.
\]
Since \(|V(G)|=|X|+|Y|\), the same bound can be written as
\[
|E(G)|\le 3|X|+2|Y|-6,
\]
and this upper bound is tight for every feasible pair \((|X|,|Y|)\) [2507.19762].

This result sharply improves the density estimate known for general bipartite \(1\)-planar graphs. Huang–Ouyang–Dong proved that every bipartite \(1\)-planar graph with partite sets \(X\) and \(Y\), \(2\le |X|\le |Y|\), satisfies
\[
|E(G)|\le 2|V(G)|+4|X|-12,
\]
and that bound is tight in the unrestricted bipartite \(1\)-planar setting. By contrast, the \(1\)-disk \(\mathcal O_X\) constraint reduces the \(|X|\)-term from \(4|X|\) to \(|X|\) [2507.19762].

Placed against other classical bounds, the class is structurally sparse but not planar-sparse. Planar graphs on \(n\ge 3\) vertices have at most \(3n-6\) edges, bipartite planar graphs have at most \(2n-4\) edges, and general \(1\)-planar graphs have at most \(4n-8\) edges. Karpov’s bound for bipartite \(1\)-planar graphs is \(3n-8\) for even \(n\ne 6\), and \(3n-9\) for odd \(n\) and for \(n=6\). For \(1\)-disk \(\mathcal O_X\) drawings, the sharper formula implies
\[
|E(G)|\le 2n+|X|-6\le \frac{5}{2}n-6
\]
because \(|X|\le n/2\) [2507.19762]. This suggests that the boundary constraint imposes a substantial combinatorial penalty relative to unrestricted bipartite \(1\)-planarity.

## 3. Extremal constructions

Tightness is established by an explicit family of drawings. For the base case \(|Y|=3(|X|-2)\), the construction starts with a maximal outerplanar graph \(H\) on \(|X|\) vertices. Such a graph has exactly \(|X|-2\) triangular faces. Into each triangular face, one inserts a configuration \(B_3\), then deletes all original edges of \(H\). The resulting graph \(G\) has \(|X|\) “black” vertices on the boundary and \(3(|X|-2)\) “blue” vertices in the interior; the blue vertices are independent and each blue vertex is adjacent to three black vertices [2507.19762].

The counting is exact. Each triangular face contributes three blue vertices and nine blue edges, so
\[
|Y|=3(|X|-2),\qquad |E(G)|=9(|X|-2).
\]
With \(|V(G)|=|X|+|Y|=4|X|-6\), this yields
\[
|E(G)|=3|X|+2|Y|-6=2|V(G)|+|X|-6.
\]
Hence the upper bound is attained in this parameter regime [2507.19762].

The general case \(|Y|=3(|X|-2)+t\), \(t\ge 1\), is obtained by adding \(t\) further interior vertices into an arbitrary region incident to two black boundary vertices and joining each new vertex to those two black vertices without any edge crossing. This preserves bipartiteness, preserves the \(1\)-disk \(\mathcal O_X\) property, and adds exactly \(t\) vertices and \(2t\) edges, so equality remains
\[
|E(G)|=2|V(G)|+|X|-6
\]
for every pair \(2\le |X|\le |Y|\) [2507.19762].

The construction is notable for its use of maximal outerplanar graphs as a boundary skeleton. Their triangulated outer-face structure creates \(|X|-2\) triangular “slots,” each of which hosts one local \(B_3\) gadget. The coefficient pattern in the extremal formula, \(3|X|+2|Y|-6\), is reflected directly in this face-by-face packing mechanism [2507.19762].

## 4. Proof strategy

The upper bound is proved by a doubling-and-gluing argument that reduces the \(1\)-disk problem to a known theorem on general bipartite \(1\)-planar graphs. Start with a \(1\)-disk \(\mathcal O_X\) drawing \(D\) of \(G\), with all \(X\)-vertices on the boundary and all \(Y\)-vertices and edges inside the disk. Perform a spherical mapping to obtain another \(1\)-planar drawing \(D'\) of the same graph in which the \(X\)-vertices still lie on the boundary of \(\mathcal O\), but all \(Y\)-vertices and edges lie in the exterior of \(\mathcal O\). The unbounded face of \(D\) becomes a bounded face \(f'\) of \(D'\) [2507.19762].

Next, place the original drawing \(D\) into the face \(f'\) of \(D'\) and identify the two boundary copies of \(X\). The resulting graph \(G^*\) is bipartite and \(1\)-planar, with
\[
|V(G^*)|=|X|+2|Y|,\qquad |E(G^*)|=2|E(G)|.
\]
Applying the Huang–Ouyang–Dong bound
\[
|E(G^*)|\le 2|V(G^*)|+4|X|-12
\]
then gives
\[
2|E(G)|\le 6|X|+4|Y|-12,
\]
hence
\[
|E(G)|\le 3|X|+2|Y|-6=2|V(G)|+|X|-6.
\]
The proof is short because the geometric constraint is encoded globally by the gluing step rather than by local case analysis [2507.19762].

Historically, Huang–Ouyang–Dong had already proved that if \(|X|=3\), then any bipartite graph with a \(1\)-disk \(\mathcal O_X\) drawing satisfies \(|E(G)|\le 2|Y|+3\), and they asked whether
\[
|E(G)|\le 2|Y|+\frac{5}{3}|X|-2
\]
holds in general. The exact theorem \(|E(G)|\le 2|V(G)|+|X|-6\) is stronger and resolves that problem completely [2507.19762].

## 5. Relation to other disk-based drawing models

Several nearby graph-drawing models also use disks or a single bounded region, but they are formally different.

| Model | Vertex placement | Edge/routing rule |
|---|---|---|
| \(1\)-disk \(\mathcal O_X\) drawing | \(X\) on one disk boundary; \(Y\) inside | Bipartite and \(1\)-planar [2507.19762] |
| Outerplanar strict confluent drawing | All vertices on one disk boundary | Edges are unique smooth paths through arcs and junctions [1308.6824] |
| Disk arrangement drawing of clustered graphs | Vertices inside cluster disks | Inter-cluster edges routed through pipes [1811.00785] |
| Disk-link drawing | Each vertex is an open disk of radius \(\rho\) | Each edge is a straight segment between disk centers [2005.02082] |
| Outer-\(1\)-planar orthogonal bar-drawing | All vertices on the outer face as bars | At most two bends per edge, embedding preserved [2009.07106] |

Outerplanar strict confluent drawings are the closest single-disk analogue in topological form, but they place **all** vertices on the boundary and interpret adjacency via unique smooth paths through a track system rather than via ordinary \(1\)-planar arcs. Clustered-graph drawings on disk arrangements use multiple disks and a pipe model for inter-cluster routing. Disk-link drawings replace point vertices by open disks with radius \(\rho\) and require each edge to be a straight segment between centers, with no non-incident disk intersected by an edge [1308.6824; 1811.00785; 2005.02082].

A different nearby literature studies outer-\(1\)-planar graphs with all vertices on the outer face. In that setting, planar visibility representations can require \(\Omega(n^2)\) area, while orthogonal bar-drawings with crossings and at most two bends per edge can achieve \(O(n\log n)\) area [2009.07106]. This is related in spirit but not equivalent: \(1\)-disk \(\mathcal O_X\) drawings constrain only one partite set to the boundary, not the entire vertex set.

Another separate line studies arrangements of prescribed disks on equally spaced rays from the origin and proves that a greedy strategy yields a covering disk of radius at most \(2\rho\), with applications to unordered tree drawings with perfect angular resolution [1109.1705]. Despite the shared “one-disk” language, that model concerns geometric disk placement and tree embedding, not bipartite \(1\)-planarity.

## 6. Significance and directions

The significance of the \(1\)-disk \(\mathcal O_X\) model lies in the way a single global topological constraint changes the extremal combinatorics of \(1\)-planar graphs. General bipartite \(1\)-planar graphs allow the tight bound \(2|V(G)|+4|X|-12\), whereas the disk-boundary restriction lowers this to \(2|V(G)|+|X|-6\) [2507.19762]. This suggests that forcing one partite set onto a single boundary component imposes a much stronger structural discipline than bipartiteness and \(1\)-planarity alone.

The extremal constructions also indicate why the class remains nontrivial. The boundary vertices of \(X\) serve as a fixed outer scaffold, while interior vertices in \(Y\) are packed into triangular regions created by a maximal outerplanar skeleton. In the sharp examples, each interior gadget contributes a controlled amount of \(1\)-planar density without violating the single-disk boundary condition [2507.19762].

The note solving the extremal problem does not explicitly list new open problems. A plausible continuation is the study of recognition, structural characterization, and extensions in which the disk constraint is modified: for example, multi-disk analogues, \(k\)-planar variants, or more general classes of boundary-constrained bipartite drawings. Another plausible direction is to compare the exact extremal behavior of \(1\)-disk \(\mathcal O_X\) drawings with the algorithmic and geometric constraints known for clustered disk arrangements, disk-link drawings, and outer-\(1\)-planar visibility models [1811.00785; 2005.02082; 2009.07106].

Source: https://www.emergentmind.com/topics/1-disk-ox-drawing