Maximum Uncrossed Subgraph Number

• The maximum uncrossed subgraph number is defined as the maximum number of uncrossed edges in a single drawing of a graph, serving as the complement to the edge crossing number.
• Analysis of h(G) uses planar subdrawings and face structures to derive tight bounds and links it to the uncrossed number and classical covering parameters like thickness.
• The parameter is pivotal in graph drawing and fixed-order book embeddings, with related decision problems being NP-complete for multi-page scenarios.

The maximum uncrossed subgraph number is a graph-drawing parameter that measures, for a single drawing of a graph $G$ in the plane, how many edges can be made uncrossed simultaneously. In the notation used in recent work, it is denoted $h(G)$ and defined as the maximum number of edges of $G$ that are not crossed in a drawing of $G$ in the plane (Charvy et al., 28 Jul 2025). This parameter is complementary to the edge crossing number, which minimizes the number of edges that are crossed at least once, and it is closely linked to the uncrossed number $\mathrm{unc}(G)$, which asks for the minimum number of drawings needed so that every edge of $G$ is uncrossed in at least one drawing (Balko et al., 2024). A distinct but related meaning also occurs in fixed-order book embeddings, where the objective is the maximum number of edges that can be kept while partitioning them into $k$ noncrossing pages under a prescribed vertex order (Jonsson et al., 2015).

1. Definitions and parameter landscape

An uncrossed collection of drawings of a graph $G=(V,E)$ is a collection $\mathcal{D}=\{D_1,\dots,D_k\}$ such that for every edge $e\in E$, there exists some drawing $h(G)$0 in which $h(G)$1 is not crossed (Hliněný et al., 2023). The uncrossed number $h(G)$2 is the least cardinality $h(G)$3 of such a collection (Balko et al., 2024). This shifts attention from a single optimal drawing to a multi-view representation in which each edge appears cleanly at least once.

Within this framework, the maximum uncrossed subgraph number is the one-drawing parameter

$h(G)$4

where the maximum ranges over drawings $h(G)$5 of $h(G)$6 in the plane (Balko et al., 2024). If $h(G)$7, then the complement $h(G)$8 is the minimum possible number of crossed edges in a drawing, i.e. the edge crossing number viewpoint (Balko et al., 2024). The two perspectives are therefore equivalent at the level of a single drawing: maximizing uncrossed edges is the same optimization as minimizing crossed edges.

The relation between $h(G)$9 and $G$0 is immediate. Since one drawing can contribute at most $G$1 uncrossed edges, any uncrossed collection must satisfy

$G$2

(Charvy et al., 28 Jul 2025). The same literature places $G$3 between classical covering parameters: $G$4 where $G$5 is thickness and $G$6 is outerthickness (Hliněný et al., 2023). This does not directly determine $G$7, but it situates the parameter inside the broader theory of planar and outerplanar edge decompositions.

A useful structural notion is that of an uncrossed subdrawing. In a drawing $G$8, the edges that are uncrossed form a planar subdrawing, and the cited structural lemma states that each edge of the original graph has its endpoints on a common face of that subdrawing (Balko et al., 2024). This face-incidence property is stronger than mere planarity and is central to the best known general bounds on $G$9.

2. General bounds for $G$0

For a connected graph $G$1 with $G$2 vertices and $G$3 edges, the strongest general bound stated directly in terms of the maximum uncrossed subgraph number is

$G$4

(Charvy et al., 28 Jul 2025). This is the main theorem of the 2025 bound paper and is presented there as the central result in the $G$5-language. Through the inequality $G$6, it yields the corresponding lower bound

$G$7

for connected graphs (Charvy et al., 28 Jul 2025).

For dense graphs with

$G$8

the same theorem specializes to the asymptotic estimate

$G$9

(Charvy et al., 28 Jul 2025). The coefficient

$\mathrm{unc}(G)$0

also appears in the dense-graph form of the uncrossed-number bound. As $\mathrm{unc}(G)$1 increases, the upper bound on the number of simultaneously uncrossed edges decreases linearly in $\mathrm{unc}(G)$2, quantifying the fact that in dense graphs only a vanishing fraction of the total $\mathrm{unc}(G)$3 edges can remain uncrossed in any one drawing.

These results refine earlier general estimates. A trivial lower bound on $\mathrm{unc}(G)$4 comes from thickness: $\mathrm{unc}(G)$5 because any planar subgraph on $\mathrm{unc}(G)$6 vertices has at most $\mathrm{unc}(G)$7 edges (Charvy et al., 28 Jul 2025). An earlier general lower bound for connected graphs was stated in the form

$\mathrm{unc}(G)$8

which, for dense graphs, was summarized as

$\mathrm{unc}(G)$9

(Charvy et al., 28 Jul 2025). By the inequality $G$0, this earlier result implicitly gives a bound of the form $G$1 up to lower-order terms, whereas the newer bound improves the dense-regime constant to $G$2 (Charvy et al., 28 Jul 2025).

The same paper also states triangle-free analogues of these estimates, indicating that forbidding triangles sharpens the face-counting argument and improves the constants for special graph classes (Charvy et al., 28 Jul 2025).

3. Proof ideas and structural method

The proof strategy for the bound on $G$3 is based on choosing an uncrossed subdrawing $G$4 of a drawing $G$5 and analyzing the planar combinatorics of $G$6 (Charvy et al., 28 Jul 2025). Because $G$7 is planar, its faces control where the remaining crossed edges of $G$8 can be routed: every edge of $G$9 has both endpoints on a common face of $k$0 (Balko et al., 2024). This converts the problem into a constrained counting question about how many additional edges can be “supported” by the faces of a planar subdrawing.

The analysis uses the face set $k$1, the face lengths $k$2, the number $k$3 of vertices on a face, and the numbers $k$4 of faces of length $k$5 (Charvy et al., 28 Jul 2025). Standard planar identities enter at this point. In particular, a planar drawing satisfies

$k$6

where $k$7 is the number of edges in the uncrossed subdrawing, and Euler’s formula gives

$k$8

for connected planar subdrawings (Charvy et al., 28 Jul 2025). Additional inequalities on weighted sums of face counts then restrict how many crossed edges can be associated with short faces.

The paper derives two complementary estimates: a “simple bound,” obtained from Euler’s formula and bounded face lengths, and a “complex bound,” obtained from a more precise accounting of how many extra edges can lie inside each face (Charvy et al., 28 Jul 2025). These are combined through a parameter $k$9, then optimized by choosing $G=(V,E)$0 as a function of the graph density. The final step selects

$G=(V,E)$1

to obtain the closed-form upper bound

$G=(V,E)$2

(Charvy et al., 28 Jul 2025). Conceptually, the argument is a refinement of the naive observation “uncrossed edges form a planar graph”: it uses not only planarity, but also the detailed face structure forced by uncrossedness.

4. Tightness, extremal constructions, and dense-regime behavior

The bound

$G=(V,E)$3

is exact for planar triangulations. If $G=(V,E)$4 is a planar triangulated graph with $G=(V,E)$5 vertices and $G=(V,E)$6 edges, then $G=(V,E)$7 has a drawing with all edges uncrossed, so $G=(V,E)$8, and substituting $G=(V,E)$9 into the bound gives equality (Charvy et al., 28 Jul 2025).

More significantly, the 2025 paper provides an asymptotically tight construction for arbitrary dense regimes below the complete-graph limit. It states that for any density

$\mathcal{D}=\{D_1,\dots,D_k\}$0

there exists a connected graph $\mathcal{D}=\{D_1,\dots,D_k\}$1 on $\mathcal{D}=\{D_1,\dots,D_k\}$2 vertices and $\mathcal{D}=\{D_1,\dots,D_k\}$3 edges such that

$\mathcal{D}=\{D_1,\dots,D_k\}$4

and

$\mathcal{D}=\{D_1,\dots,D_k\}$5

(Charvy et al., 28 Jul 2025). Since the difference between the general upper bound and this lower bound is at most

$\mathcal{D}=\{D_1,\dots,D_k\}$6

the theorem shows that the upper bound is tight up to additive lower-order terms on dense graphs for all fixed $\mathcal{D}=\{D_1,\dots,D_k\}$7 below $\mathcal{D}=\{D_1,\dots,D_k\}$8 (Charvy et al., 28 Jul 2025).

The construction begins with a wheel $\mathcal{D}=\{D_1,\dots,D_k\}$9, drawn planarly, and then adds all possible edges between the outer-cycle vertices to form a clique $e\in E$0 inside the outer face, with those added edges crossing each other (Charvy et al., 28 Jul 2025). The remaining $e\in E$1 vertices are inserted one by one into triangular faces of the planar part and connected to the three vertices of the chosen triangle, thereby preserving a triangulated interior. In the notation stated in the construction,

$e\in E$2

and

$e\in E$3

Since $e\in E$4, the construction yields

$e\in E$5

(Charvy et al., 28 Jul 2025).

For complete graphs, the dense-regime asymptotics become especially transparent. The paper notes that the general bound is tight up to low-order terms for $e\in E$6, “as warranted by complete graphs” (Charvy et al., 28 Jul 2025). This matches the exact value

$e\in E$7

which is the complete-graph case of the classical one-drawing maximum (Balko et al., 2024). Since $e\in E$8 has density asymptotically $e\in E$9, the coefficient $h(G)$00 becomes $h(G)$01, agreeing with the leading term of $h(G)$02.

5. Exact values for classical graph families

The exact value

$h(G)$03

for complete graphs plays a central role in the theory (Balko et al., 2024). It supplies both a benchmark for tightness and the lower-bound ingredient used in exact uncrossed-number formulas for complete graphs. The corresponding uncrossed number is

$h(G)$04

showing how the one-drawing constraint encoded by $h(G)$05 propagates to the multi-drawing covering problem (Balko et al., 2024).

For complete bipartite graphs, Mengersen’s exact determination of the one-drawing parameter is

$h(G)$06

(Balko et al., 2024). The exact uncrossed numbers are then

$h(G)$07

These formulas show that the structural shape of a graph class can determine $h(G)$08 exactly, and that the transition between density regimes $h(G)$09 and $h(G)$10 already appears at the level of the maximum number of simultaneously uncrossed edges (Balko et al., 2024).

A broader implication is that $h(G)$11 is not merely an auxiliary bound for $h(G)$12; in several classical families it is the decisive parameter from which the exact uncrossed number is recovered by refined counting and structural arguments (Balko et al., 2024).

6. Algorithmic status and the fixed-order book-embedding variant

From the single-drawing viewpoint, deciding whether a graph has a drawing with at most $h(G)$13 crossed edges is the edge crossing number problem, and the complementary formulation asks whether there is a drawing with at least $h(G)$14 uncrossed edges (Balko et al., 2024). The paper proves that the edge crossing number problem is NP-complete, and by complementarity the “maximum uncrossed edges” decision problem is NP-complete as well (Balko et al., 2024). The 2025 survey paper also notes an NP-completeness result for computing $h(G)$15 and mentions an FPT algorithm of Colin de Verdière and Hliněný parameterized by the number of crossed edges (Charvy et al., 28 Jul 2025). This places the parameter in the familiar pattern of graph-drawing optimization: hard in general, but structurally tractable under suitable parameterization.

A different use of the phrase “maximum uncrossed subgraph number” arises in fixed-order book embeddings. Given a graph $h(G)$16, a total order $h(G)$17 on its vertices, and an integer $h(G)$18, one may define

$h(G)$19

that is, the maximum number of edges that can be retained while assigning them to $h(G)$20 pages so that edges on the same page are pairwise noncrossing with respect to the fixed spine order (Jonsson et al., 2015). The corresponding decision problem is exactly the Maximum Pagenumber-$h(G)$21 Subgraph problem.

In this fixed-order model there is a sharp complexity dichotomy. For $h(G)$22, the problem can be solved in time

$h(G)$23

using dynamic programming (Jonsson et al., 2015). For every fixed $h(G)$24, however, the decision problem is NP-complete (Jonsson et al., 2015). The same paper proves NP-hardness, and hence NP-completeness, for the acyclic directed variant as well (Jonsson et al., 2015). Structurally, the difference is that a one-page embedding is simply a single noncrossing layer under the given order, whereas two or more pages allow several uncrossed layers whose interaction already encodes NP-hardness.

This fixed-order notion is not identical to the plane-drawing parameter $h(G)$25, because it is constrained by a prescribed vertex order and a bounded page budget. Nonetheless, both parameters quantify the same underlying resource: how large a subgraph can remain uncrossed under a specified drawing model. The modern literature therefore uses “maximum uncrossed subgraph number” both for the one-drawing planar parameter $h(G)$26 and, in an order-constrained setting, for the optimization value of maximum pagenumber-$h(G)$27 subgraph problems (Charvy et al., 28 Jul 2025, Jonsson et al., 2015).