Octilinear Realizability in Graph Drawing

• Octilinear realizability is a graph drawing model that restricts edge segments to eight compass directions, integrating prescribed embeddings and bend constraints.
• It employs port assignments, angle sequences, and bend specifications to decide if a planar graph can be drawn without unwanted intersections, with NP-hard cases highlighting complex geometric reductions.
• Efficient algorithms arise under fixed-parameter settings, and comparisons with orthogonal models reveal trade-offs in edge-complexity and area requirements.

Octilinear realizability concerns the existence of planar octilinear drawings for a given abstract planar graph together with a prescribed combinatorial embedding and geometric constraints, such as port assignments (directions of edge incidence), angles at vertices, and precise edge-bend specifications. The octilinear drawing model restricts edge segments to the eight compass slopes: 0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°. Octilinear representations arise in applications such as metro-map schematization, where readability benefits from a small set of uniformly angled edge segments.

1. Octilinear Drawing Model and Problem Formalization

An octilinear drawing of a planar graph $G=(V,E)$ embeds each vertex $v\in V$ at a unique point in the plane, with each edge $uv\in E$ realized as a sequence of straight-line segments, each aligned with an octilinear direction ({0°, 45°, ..., 315°}). Each vertex can support up to eight incident edges ("ports"), so the maximum degree supported is 8, though most theoretical results are for lower-degree graphs ($\Delta\le 4$ or 5). An octilinear representation $\mathcal{R}$ of $G$ prescribes:

• The planar embedding: the cyclic order of edges around every vertex and a choice of the external face;
• For each incidence, the direction (port) used by the edge;
• At each vertex, the sequence of angles between consecutive incident edges (multiples of 45° from the set $\{45^\circ,90^\circ,...,315^\circ\}$);
• For each edge, the required number and type of bends, with each bend being a $\pm45^\circ$ turn and specified in order from one endpoint to the other.

A drawing $\Gamma$ realizes $\mathcal{R}$ if edges and bends match the prescribed direction and angles, and intersections occur only at shared endpoints. Octilinear Realizability asks whether such a drawing exists for a given $\mathcal{R}$ (Bekos et al., 2017, Förster et al., 25 Nov 2025).

2. Complexity and Hardness Results

Octilinear realizability is computationally intractable in key settings. For the bendless case (edge-complexity one), the following fixed-representation decision problem is central:

• Input: Planar graph $G$ (typically with $\Delta\le 4$), and an octilinear representation $R$ fixing both embedding and all port assignments.
• Question: Does a planar, bendless octilinear drawing of $G$ exist that realizes $R$ exactly?

Bekos et al. show that this problem is NP-hard, even for maximum degree $\Delta\le 4$ (Bekos et al., 2017). The reduction is from 3-SAT and encodes the satisfiability of a Boolean formula using the lengths of certain edges in the octilinear representation, enforced through complex geometric gadgets. The construction uses degree-4 gadgets such as unit-length edges, variable gadgets (which force a relation $\ell(x) + \ell(\neg x) = 3\ell(u)$), parity gadgets (ensuring binary assignments), clause gadgets (forcing coverage of clauses only if at least one literal is true), along with crossing and copy gadgets. This encoding ensures that the existence of a solution corresponds exactly to the satisfiability of the original formula.

Recent results extend this hardness to more general settings: octilinear realizability remains NP-hard even if the representation contains only one non-convex face ($\varphi=1$), or if each interior face has at most eight reflex corners ($\kappa=8$) (Förster et al., 25 Nov 2025). These results demonstrate that the intractability is robust even under strong geometric constraints, distinguishing octilinear realizability from its orthogonal counterpart, which is solvable in linear time.

3. Parameterized and Positive Results

A notable algorithmic result is that octilinear realizability becomes fixed-parameter tractable (FPT) when parameterized by the total number of reflex corners $\omega$ in the representation. The approach involves:

• Reducing the input to an equivalent instance with all non-convex faces of size $O(\omega)$ via "shadow graphs" and local expansions.
• For each possible alignment of the reflex corners, reducing to a convex-only instance solvable by network flows—one for horizontal and one for vertical dimensions, coupled through diagonals—mirroring flow-conservation conditions among edge lengths and face constraints.
• Enumerating all alignment choices gives a total of $2^{O(\omega^2)}$ options, each checkable in $n^{O(1)}$ time.

Thus, octilinear realizability can be solved in $2^{O(\omega^2)}n^{O(1)}$ time (Förster et al., 25 Nov 2025). For instances with no internal reflex corners and at most four on the outer face, this yields efficient algorithms for both realizability and (via a skeleton reduction plus exhaustive search) for the compaction problem—finding a minimum-area integer grid drawing.

4. Edge-Complexity, Degree Constraints, and Constructive Bounds

The number of bends per edge (edge-complexity) is a fundamental aspect of octilinear realizability. For $k$-planar graphs (maximum degree $k$), the following hold:

Degree Bound	Edge-Complexity	Area Requirement	Reference
4-planar	$\leq 1$	$O(n^2)\times O(n)$	(Bekos et al., 2014)
5-planar	$\leq 1$	Super-polynomial	(Bekos et al., 2014)
6-planar (some)	$\geq 2$ (necessary)	–	(Bekos et al., 2014)

Every 4-planar graph admits a planar octilinear drawing with at most one bend per edge realizable on an $O(n^2)\times O(n)$ grid, computable in $O(n)$ time for triconnected input (Bekos et al., 2014). For 5-planar graphs, the one-bend-per-edge result holds but the area may be exponential. There are infinite families of 6-planar graphs requiring at least two bends per edge, shown via recursive port saturation arguments.

Canonical orderings (for triconnected graphs), SPQR-trees (for biconnected graphs), and block-cut trees (for general connected graphs) underpin efficient constructive algorithms for low-degree cases, with invariants tailored to maintain port assignments, stretchability, and planarity. The progress for higher degrees is strictly limited by edge-complexity lower bounds rooted in "port saturation" phenomena.

5. Comparison to Orthogonal and Smooth-Orthogonal Models

Octilinear models extend the classic orthogonal drawings (which use only horizontal and vertical segments) by allowing diagonal ($\pm45^\circ$) segments, expanding the set of possible embeddings and correspondences. The classes of graphs admitting bendless (edge-complexity 1) drawings in the octilinear and smooth-orthogonal models are incomparable:

• There exist infinite families of planar graphs in smooth-orthogonal edge-complexity-1 (SC$_1$) but not in octilinear edge-complexity-1 (8C$_1$), and vice versa (Bekos et al., 2017).
• All planar graphs with $\Delta\le4$ are in smooth-orthogonal edge-complexity-2 (SC$_2$).
• Octilinear edge-complexity-2 (8C$_2$) strictly contains SC$_2$ but is strictly contained in 8C$_3$ (all planar graphs with $\Delta \leq 8$).

This non-nesting reflects the distinct geometric and combinatorial properties—most notably that additional slopes allow certain embeddings not possible in orthogonal realizations, but also create new constraints and hardness phenomena.

6. Open Questions and Research Directions

Several open problems remain in the paper of octilinear realizability:

• Whether octilinear realizability is polynomial-time solvable for fixed-face reflex-corner bounds $\kappa < 8$, or if NP-hardness persists for all nontrivial $\kappa$ (Förster et al., 25 Nov 2025).
• Whether the $2^{O(\omega^2)}$ parameterized complexity can be improved to $c^\omega$ for some constant $c$, or if a polynomial kernel exists.
• Questions of area minimization: whether 4-planar (resp. 5-planar) graphs can be drawn with one bend per edge in subcubic or polynomial area, and the complexity of the compaction problem in the octilinear setting (Bekos et al., 2014).
• Whether every triangle-free 6-planar graph admits a 1-bend octilinear drawing, and the recognition complexity for 1-bend realizability in higher-degree classes.

A plausible implication is that new combinatorial and geometric techniques will be needed to bypass the current lower bound phenomena, particularly regarding area requirements and efficient recognition under structural assumptions.

7. Applications and Broader Impact

Octilinear realizability underpins layout algorithms for schematic maps (e.g., public transport networks), information visualization, and VLSI design, where edge complexity, angle restrictions, and readability are fundamental. The juxtaposition of intractability in the realizability problem with available efficient algorithms in specific parameter regimes (e.g., low total non-convexity or degree) suggests ongoing interest in both structural graph theory and practical geometry-driven computation.

The connections between orthogonal, smooth-orthogonal, and octilinear models highlight the intricate balance between geometric flexibility and algorithmic tractability, with octilinear realizability offering a revealing lens on the trade-offs present in constraint-based graph drawing (Bekos et al., 2017, Förster et al., 25 Nov 2025, Bekos et al., 2014).