Ordered Level Planarity

• Ordered Level Planarity is a graph drawing model where vertices are assigned fixed levels and prescribed left-to-right positions, enforcing strict planar y-monotone curves.
• The problem is NP-complete even with maximum degree 2 and level-width 2, while specific cases such as single-vertex levels or proper instances are solvable in linear time.
• The rigid ordering not only complicates the drawing process but also underpins the computational hardness of related problems like Geodesic, Manhattan, and Bi-Monotonicity planarity.

Searching arXiv for the target paper and closely related graph-drawing context. {} Ordered Level Planarity is the fully position-constrained analogue of Level Planarity: vertices are assigned both levels and prescribed left-to-right positions, and every edge must be drawn as a planar $y$-monotone curve. In this formulation, the ordered constraint is not a cosmetic refinement of ordinary level drawing; it changes the complexity landscape decisively. The problem is $\mathcal{NP}$-complete even for instances with maximum degree $\Delta=2$ and level-width $\lambda=2$, while several sharply delimited subclasses remain linear-time solvable. The same ordered rigidity also serves as a hardness source for Geodesic Planarity, Manhattan Geodesic Planarity, Bi-Monotonicity, T-Level Planarity, Clustered Level Planarity, and Constrained Level Planarity (Klemz et al., 2017).

1. Formal model and the ordered constraint

A level graph is a directed graph

$\mathcal G=(G,\gamma), \qquad G=(V,E),$

with a surjective level assignment

$\gamma:V\rightarrow \{0,\dots,h\},$

such that

$$\gamma(u)<\gamma(v)\quad \text{for every edge }(u,v)\in E.$$

The $i$-th level is

$V_i=\{v\mid \gamma(v)=i\},$

its width is

$\lambda_i=|V_i|,$

and the level-width is

$\mathcal{NP}$0

A level planar drawing places each vertex $\mathcal{NP}$1 on the horizontal line

$\mathcal{NP}$2

with $\mathcal{NP}$3, and draws each edge $\mathcal{NP}$4 as a $\mathcal{NP}$5-monotone curve from $\mathcal{NP}$6 to $\mathcal{NP}$7 without crossings (Klemz et al., 2017).

Ordered Level Planarity adds a prescribed left-to-right position on each level. An ordered level graph is

$\mathcal{NP}$8

where $\mathcal{NP}$9 is a level graph and

$\Delta=2$0

is a level ordering such that, for each level $\Delta=2$1, the restriction of $\Delta=2$2 to $\Delta=2$3 is a bijection onto

$\Delta=2$4

An ordered level planar drawing is then a level planar drawing in which every vertex $\Delta=2$5 is placed at the prescribed coordinates

$\Delta=2$6

This is the essential distinction from standard Level Planarity. Ordinary Level Planarity fixes only the $\Delta=2$7-coordinates via $\Delta=2$8; Ordered Level Planarity fixes both the $\Delta=2$9-coordinate through $\lambda=2$0 and the left-to-right order, equivalently the $\lambda=2$1-coordinate, through $\lambda=2$2. The paper stresses that $\lambda=2$3 and $\lambda=2$4 are merely encodings of total and partial orders: the problem is unchanged if one lets them map to arbitrary reals, or if one requires only that vertices on each level appear in the prescribed total order rather than at exact integer coordinates. A level graph is called proper if

$\lambda=2$5

2. Complexity dichotomy

The central theorem is a sharp dichotomy in terms of maximum degree and level-width. Ordered Level Planarity is $\lambda=2$6-complete even for maximum degree

$\lambda=2$7

and level-width

$\lambda=2$8

By contrast, three regimes are linear-time solvable: the case $\lambda=2$9, the case

$\mathcal G=(G,\gamma), \qquad G=(V,E),$0

and proper instances (Klemz et al., 2017).

The hard side is unusually strong. The first reduction yields instances of degree at most $\mathcal G=(G,\gamma), \qquad G=(V,E),$1, and a separate width-reduction lemma converts them to width $\mathcal G=(G,\gamma), \qquad G=(V,E),$2. The construction also has a more refined property: all vertices on levels of width at least $\mathcal G=(G,\gamma), \qquad G=(V,E),$3 can be arranged to have out-degree at most $\mathcal G=(G,\gamma), \qquad G=(V,E),$4 and in-degree at most $\mathcal G=(G,\gamma), \qquad G=(V,E),$5. Thus hardness does not arise from high local branching or from wide levels alone; it survives after both parameters are tightly constrained.

The easy side isolates exactly those situations where the prescribed order does not create enough combinatorial rigidity to encode crossings or choices. If $\mathcal G=(G,\gamma), \qquad G=(V,E),$6, each level contains only one vertex, so Ordered Level Planarity collapses to Level Planarity on a graph with one vertex per level. If

$\mathcal G=(G,\gamma), \qquad G=(V,E),$7

the graph is a disjoint union of $\mathcal G=(G,\gamma), \qquad G=(V,E),$8-monotone directed paths, and the remaining problem is only to decide a consistent global left-to-right ordering of those paths. If the instance is proper, all interactions are local between consecutive levels, and a sweep through the levels suffices.

These results identify the negative corner of the landscape precisely: prescribing a total order on each level is algorithmically harmless only under strong structural restrictions, while width $\mathcal G=(G,\gamma), \qquad G=(V,E),$9 and degree $\gamma:V\rightarrow \{0,\dots,h\},$0 already suffice for $\gamma:V\rightarrow \{0,\dots,h\},$1-completeness.

3. NP-completeness proof architecture

The hardness reduction is from Planar Monotone 3-Satisfiability. The source instance is given by a monotone 3SAT formula together with a monotone rectilinear representation $\gamma:V\rightarrow \{0,\dots,h\},$2 of the variable-clause graph: variables are horizontal segments on a line $\gamma:V\rightarrow \{0,\dots,h\},$3, positive clauses lie above $\gamma:V\rightarrow \{0,\dots,h\},$4, negative clauses lie below $\gamma:V\rightarrow \{0,\dots,h\},$5, and clauses are represented by rotated E-shapes. From such an instance $\gamma:V\rightarrow \{0,\dots,h\},$6, the construction builds an ordered level graph

$\gamma:V\rightarrow \{0,\dots,h\},$7

with four tiers of levels,

$\gamma:V\rightarrow \{0,\dots,h\},$8

Each clause $\gamma:V\rightarrow \{0,\dots,h\},$9 is represented by a clause edge

$$\gamma(u)<\gamma(v)\quad \text{for every edge }(u,v)\in E.$$0

from tier $$\gamma(u)<\gamma(v)\quad \text{for every edge }(u,v)\in E.$$1 to tier $$\gamma(u)<\gamma(v)\quad \text{for every edge }(u,v)\in E.$$2. The geometric idea is a rigid system of tunnels: each clause edge must pass through one tunnel corresponding to one literal of its clause, while each variable gadget blocks either the positive or the negative tunnel family for that variable (Klemz et al., 2017).

Tier $$\gamma(u)<\gamma(v)\quad \text{for every edge }(u,v)\in E.$$3 contains clause gadgets. Each gadget has $$\gamma(u)<\gamma(v)\quad \text{for every edge }(u,v)\in E.$$4 vertices and three distinguished segments, the gates

$$\gamma(u)<\gamma(v)\quad \text{for every edge }(u,v)\in E.$$5

The gadget, without its clause edge, has a unique ordered level planar drawing in the sense that on every level the left-to-right sequence of vertices and intersected edges is fixed. Because of this rigidity, the clause edge must pass through exactly one of the three gates, corresponding to one of the clause’s literals. This is the first decisive use of the ordered constraint: the prescribed order forces a unique combinatorial layout.

Tier $$\gamma(u)<\gamma(v)\quad \text{for every edge }(u,v)\in E.$$6 merges gates belonging to the same literal into literal tunnels. For variable $$\gamma(u)<\gamma(v)\quad \text{for every edge }(u,v)\in E.$$7, the paper denotes the positive and negative tunnels by

$$\gamma(u)<\gamma(v)\quad \text{for every edge }(u,v)\in E.$$8

A clause edge is drawn in such a tunnel when it lies in the region between the two paths defining that tunnel. After leaving its clause gadget, the clause edge must continue inside the chosen literal tunnel.

The variable gadgets occupy tiers $$\gamma(u)<\gamma(v)\quad \text{for every edge }(u,v)\in E.$$9 and $i$0. They are nested: they start in the order

$i$1

in $i$2 and end in reverse order in $i$3. The long edge

$i$4

must be drawn either to the left or to the right of a designated vertex $i$5 in $i$6, and this choice propagates through the construction. In one state the gadget blocks $i$7, and in the other state it blocks $i$8. The blocking is enforced by edges such as

$i$9

together with the fixed vertex order. The outcome is that for each variable exactly one corridor family remains usable: either positive clauses may route through $V_i=\{v\mid \gamma(v)=i\},$0, or negative clauses may route through $V_i=\{v\mid \gamma(v)=i\},$1, but not both.

Tier $V_i=\{v\mid \gamma(v)=i\},$2 is largely technical. It contains $V_i=\{v\mid \gamma(v)=i\},$3 levels, one terminal vertex $V_i=\{v\mid \gamma(v)=i\},$4 per level, with

$V_i=\{v\mid \gamma(v)=i\},$5

Because each such level contains only one vertex, arrivals there impose no further ordering constraints.

Correctness is bidirectional. From a drawing, one sets $V_i=\{v\mid \gamma(v)=i\},$6 if $V_i=\{v\mid \gamma(v)=i\},$7 is blocked and $V_i=\{v\mid \gamma(v)=i\},$8 otherwise. Every clause edge must choose one of its gates and then a corresponding literal tunnel, and that tunnel is usable only if the associated literal is true. Conversely, from a satisfying assignment, one draws the nested variable gadgets according to the truth values, leaves the appropriate tunnel family unblocked, and routes each clause edge through a gate and tunnel associated with a true literal. The fixed order is indispensable throughout: without it, standard Level Planarity could permute vertices on levels and destroy the tunnel-and-barrier mechanism.

4. Linear-time regimes

The tractable subclasses all arise from strong structural simplifications. When

$V_i=\{v\mid \gamma(v)=i\},$9

the graph is a set $\lambda_i=|V_i|,$0 of disjoint $\lambda_i=|V_i|,$1-monotone paths. The paper defines a precedence relation on paths: $\lambda_i=|V_i|,$2 if $\lambda_i=|V_i|,$3 have vertices $\lambda_i=|V_i|,$4 on a common level with

$\lambda_i=|V_i|,$5

This means that $\lambda_i=|V_i|,$6 must be drawn to the left of $\lambda_i=|V_i|,$7. If $\lambda_i=|V_i|,$8 is acyclic, a linear extension provides a feasible left-to-right ordering of the paths and hence a drawing; if $\lambda_i=|V_i|,$9 contains a cycle, no drawing exists. The test is executable in

$\mathcal{NP}$00

time by topological sorting (Klemz et al., 2017).

When $\mathcal{NP}$01, there is no left-right choice at all. Ordered Level Planarity reduces to ordinary Level Planarity on a graph with one vertex per level, so the case is linear-time solvable.

For proper instances, the paper states that Ordered Level Planarity is solvable in linear time by sweeping through the levels. The structural reason is localness: every edge connects consecutive levels, so the interaction pattern can be checked incrementally. This suggests that the source of hardness in the general problem is not only fixed order, but fixed order together with long-range edge interactions across multiple levels.

5. Geodesic Planarity and Bi-Monotonicity

A major consequence of the ordered framework is that it transfers hardness to fixed-position drawing problems. A Geodesic Planarity instance is

$\mathcal{NP}$02

where $\mathcal{NP}$03 prescribe vertex positions and $\mathcal{NP}$04 is a finite direction set symmetric with respect to the origin. A drawing is geodesic with respect to $\mathcal{NP}$05 if each edge is a polygonal path using two adjacent directions from $\mathcal{NP}$06. The paper proves that for every symmetric direction set $\mathcal{NP}$07 with

$\mathcal{NP}$08

Geodesic Planarity is $\mathcal{NP}$09-hard, even when the graph is a perfect matching and the prescribed coordinates are in general position (Klemz et al., 2017).

The reduction proceeds in two steps. First, an Ordered Level Planarity instance with

$\mathcal{NP}$10

is sheared by translating vertices on level $\mathcal{NP}$11 by $\mathcal{NP}$12 units to the right. This preserves the encoded left-right constraints while making every edge segment have positive slope; in the Manhattan case it can then be replaced by a nearby orthogonal $\mathcal{NP}$13-geodesic path. Second, each vertex $\mathcal{NP}$14 is split into a small gadget placed in a square $\mathcal{NP}$15 centered at

$\mathcal{NP}$16

Incident edges are attached to degree-$\mathcal{NP}$17 vertices near the square’s corners, and an additional blocking edge is inserted from top-left to bottom-right. Every non-blocking edge runs from bottom-left to top-right, so it cannot pass through another gadget square without crossing that square’s blocking edge. This localizes the geometry and turns the graph into a perfect matching.

The result applies in particular to Manhattan Geodesic Planarity, where $\mathcal{NP}$18 consists of the four horizontal and vertical directions. Katz, Krug, Rutter, and Wolff had claimed polynomial-time solvability for perfect matchings, but the hardness theorem implies that this is incorrect unless

$\mathcal{NP}$19

Bi-Monotonicity asks for a planar drawing with fixed vertex positions in which every edge is both $\mathcal{NP}$20-monotone and $\mathcal{NP}$21-monotone. The paper notes that every $\mathcal{NP}$22-geodesic rectilinear path can be transformed into a bi-monotone curve and vice versa. Hence the Manhattan reduction immediately yields: $\mathcal{NP}$23 Ordered Level Planarity is therefore not only a graph-drawing problem in its own right, but also a hardness core for several fixed-position variants.

6. Ordered Level Planarity inside broader constrained level models

Ordered Level Planarity is a very restricted special case of several broader level-drawing models, so its hardness results strengthen earlier negative results for those settings. The reduction to Constrained Level Planarity is immediate: Ordered Level Planarity is exactly the case in which the allowed order on each level is a single total order (Klemz et al., 2017).

For T-Level Planarity, each level $\mathcal{NP}$24 is equipped with a tree $\mathcal{NP}$25, and leaves in each subtree must appear consecutively. The paper reduces Ordered Level Planarity to T-Level Planarity by first ensuring width $\mathcal{NP}$26, then adding two global paths $\mathcal{NP}$27 and $\mathcal{NP}$28 intended to stay to the left and right of the original vertices, and choosing the level trees so that the designated vertex $\mathcal{NP}$29 must be consecutive with the left anchor $\mathcal{NP}$30 and $\mathcal{NP}$31 with the right anchor $\mathcal{NP}$32. Any T-level planar drawing then forces $\mathcal{NP}$33 to lie left of $\mathcal{NP}$34 on every level, thereby recovering the ordered constraint.

For Clustered Level Planarity, the reduction uses only two non-trivial clusters,

$\mathcal{NP}$35

After padding levels if necessary, the designated vertices $\mathcal{NP}$36 and $\mathcal{NP}$37 are assigned so that $\mathcal{NP}$38 contains all $\mathcal{NP}$39 and $\mathcal{NP}$40 contains all $\mathcal{NP}$41. The edges are heavily subdivided to satisfy cluster-boundary crossing rules. Because the two cluster regions pass through every level and cannot intersect or nest improperly, one may assume $\mathcal{NP}$42 is always left of $\mathcal{NP}$43, so the ordered constraint is encoded by cluster membership. This yields the consequence that Clustered Level Planarity remains $\mathcal{NP}$44-hard even for flat hierarchies with only two non-trivial clusters, answering a question posed by Angelini, Da Lozzo, Di Battista, Frati, and Roselli.

The overall significance is structural. Standard Level Planarity is polynomial-time solvable, but fixing a total order on every level removes exactly the flexibility that makes the ordinary problem tractable. In the ordered setting, levels become rigid corridors, clause gadgets become unique, tunnels become unavoidable, and variable gadgets can block whole route families. This suggests that Ordered Level Planarity is a minimal rigid form of level drawing: once the per-level order is prescribed, Boolean choice and clause satisfaction can already be encoded within degree-$\mathcal{NP}$45, width-$\mathcal{NP}$46 instances.