Constrained Incremental Graph Drawing Problem

• C-IGDP is a hierarchical graph drawing problem that updates an existing multilayer layout by incrementally adding vertices and arcs while preserving relative vertex order and displacement bounds.
• The approach employs a mixed-integer formulation and heuristic methods such as GRASP and its hybrid variant, GL-GRASP, to optimize crossing minimization efficiently.
• Empirical evaluations demonstrate that learning-enhanced strategies achieve a superior quality–time balance, particularly in denser and larger graph instances.

The Constrained Incremental Graph Drawing Problem (C-IGDP), as formalized in recent work, is a hierarchical graph visualization problem in which an original multilayer drawing is updated by inserting incremental vertices and arcs while preserving the user’s mental map through two explicit feasibility conditions: the relative order of original vertices in each layer must remain unchanged, and the absolute displacement of each original vertex is bounded by a parameter $d$. The optimization goal is to minimize the number of crossings in the resulting incremental drawing (Charytitsch et al., 21 Aug 2025). In a broader research interpretation, C-IGDP also sits within a larger family of constrained extension and redraw problems in which a pre-existing drawing, embedding, region assignment, or geometric convention restricts how new graph elements may be inserted or how an existing layout may be adjusted.

1. Canonical multilayer formulation

In the formulation studied explicitly under the name C-IGDP, the original drawing is $\mathcal{D}=(G,\Pi_0)$, where $G=(V,A,\Lambda)$ is a multilayer graph, $V^\lambda$ is the set of vertices in layer $\lambda$, $A^\lambda$ is the set of arcs leaving layer $\lambda$, and $\Lambda$ is the number of layers. The position functions are $\Pi_0=\{\pi_0^1,\ldots,\pi_0^\Lambda\}$, with $$\pi_0^\lambda:V^\lambda\mapsto\{1,\ldots,|V^\lambda|\}$$. Incremental vertices $\mathcal{D}=(G,\Pi_0)$0 and incremental arcs $\mathcal{D}=(G,\Pi_0)$1 are then added, yielding the incremental graph $\mathcal{D}=(G,\Pi_0)$2, with $\mathcal{D}=(G,\Pi_0)$3 and $\mathcal{D}=(G,\Pi_0)$4 (Charytitsch et al., 21 Aug 2025).

Arc crossings are defined layerwise. Two arcs $\mathcal{D}=(G,\Pi_0)$5 and $\mathcal{D}=(G,\Pi_0)$6, with $\mathcal{D}=(G,\Pi_0)$7 and $\mathcal{D}=(G,\Pi_0)$8, cross if the order of their tails is opposite to the order of their heads, namely either $\mathcal{D}=(G,\Pi_0)$9 and $G=(V,A,\Lambda)$0, or the symmetric alternative. The output is a new drawing $G=(V,A,\Lambda)$1, where $G=(V,A,\Lambda)$2 gives final positions (Charytitsch et al., 21 Aug 2025).

Feasibility is defined by two constraints on original vertices. First, relative precedence is preserved: for all original vertices $G=(V,A,\Lambda)$3,

$G=(V,A,\Lambda)$4

Second, absolute displacement is bounded: $G=(V,A,\Lambda)$5 This makes the incremental task neither a complete redraw nor an unconstrained crossing minimization problem. The original ordering acts as a hard combinatorial invariant, while $G=(V,A,\Lambda)$6 imposes a hard geometric tolerance on positional perturbation (Charytitsch et al., 21 Aug 2025).

The mixed-integer formulation uses binary precedence variables $G=(V,A,\Lambda)$7, crossing variables $G=(V,A,\Lambda)$8, and integer position variables $G=(V,A,\Lambda)$9. Its objective is

$V^\lambda$0

subject to crossing-definition constraints, transitivity and completeness of precedence, preservation of original relative order, displacement bounds, and the link between precedence variables and final positions: $V^\lambda$1 The unconstrained IGDP is obtained by using constraints (1)–(6) and (10), i.e., without the absolute-position restrictions (Charytitsch et al., 21 Aug 2025).

2. Heuristic and hybrid solution methods

The principal prior methods emphasized for C-IGDP are GRASP-based heuristics, especially GRASP2 and GRASP3, together with a Tabu Search. Standard GRASP is iterative: a semi-greedy construction generates a feasible solution, local search improves it, and the best solution over all iterations is retained. A restricted candidate list (RCL) contains the best $V^\lambda$2 candidates according to a greedy score, where $V^\lambda$3 controls greediness versus randomness (Charytitsch et al., 21 Aug 2025).

Three earlier construction procedures are described. In $V^\lambda$4, a high-degree vertex is first selected and placed randomly, and remaining vertices are chosen semi-greedily by degree and placed by the barycenter heuristic. In $V^\lambda$5, the original drawing is used as the initial partial solution, and only incremental vertices are inserted semi-greedily. In $V^\lambda$6, the key score is insertion cost: $V^\lambda$7 denotes the number of additional crossings caused by inserting vertex $V^\lambda$8 at position $V^\lambda$9, and $\lambda$0 is the minimum over positions. The RCL threshold is

$\lambda$1

The local search of the literature heuristics uses two neighborhoods, swap and insert, both restricted to incremental vertices and both preserving the constraints on original vertices (Charytitsch et al., 21 Aug 2025).

The hybrid method proposed specifically for C-IGDP is Graph Learning GRASP (GL-GRASP). It retains the overall GRASP structure and the same local search, but replaces the construction phase by one guided by graph representation learning. Its preprocessing step computes node embeddings for all vertices of the incremental graph $\lambda$2, projects them into $\lambda$3-D by PCA, and computes Euclidean distances between adjacent nodes in this projected space. These latent-space distances are then used as heuristic attributes during semi-greedy insertion (Charytitsch et al., 21 Aug 2025).

The node score in GL-GRASP is

$\lambda$4

where $\lambda$5 is the current partial solution. The RCL is defined by

$\lambda$6

with

$\lambda$7

Selected vertices are placed using three embedding-informed positional mechanisms: the closest available position to the nearest neighbor in an adjacent layer, the closest available position to the farthest neighbor in an adjacent layer, and the closest available position to the average of the positions already collected in the candidate set $\lambda$8. The paper interprets these choices as preserving community structure, avoiding excessive clustering, and enabling smooth intermediate placement (Charytitsch et al., 21 Aug 2025).

The embedding methods evaluated are HOPE, Spectral (SPEC), Node2Vec (N2V), and SDNE. The stochastic variants G_N2V* and G_SDNE* recompute embeddings and the distance matrix at every GRASP iteration, whereas the standard variants compute embeddings once and reuse them. The implementation uses CogDL defaults; for N2V and SDNE the default embedding dimension is $\lambda$9, while for HOPE and SPEC the dimension depends on graph size according to the CogDL implementation (Charytitsch et al., 21 Aug 2025).

3. Empirical behavior and evaluation

The experimental methodology centers on the Normalized Primal Integral (NPI), which measures solution quality over time rather than final quality alone. Let $A^\lambda$0 be the best solution value among all GL-GRASP heuristics for an instance, let $A^\lambda$1, and let $A^\lambda$2 be the maximum time among all GL-GRASP heuristics on that instance. If $A^\lambda$3 are the successive improving solution values found at times $A^\lambda$4, then

$A^\lambda$5

Lower NPI is better. This measure is central because the paper’s main claim is a quality–time improvement, not merely an improved final crossing count (Charytitsch et al., 21 Aug 2025).

The benchmark dataset contains 609 test cases derived from 240 base instances. Original hierarchical graphs have $A^\lambda$6, $A^\lambda$7, $A^\lambda$8, or $A^\lambda$9 layers, density $\lambda$0, and $\lambda$1 to $\lambda$2 non-incremental vertices per layer; the incremental ratio is $\lambda$3 or $\lambda$4, and $\lambda$5. A second scalability experiment uses denser, larger instances with $\lambda$6, density reported as $\lambda$7, $\lambda$8, $\lambda$9, and $\Lambda$0–$\Lambda$1 original nodes per layer (Charytitsch et al., 21 Aug 2025).

In the first experiment, which compares only GL-GRASP variants, G_SPEC attains the lowest mean gap, $\Lambda$2, with mean time $\Lambda$3 s, while G_SDNE attains the best mean runtime, $\Lambda$4 s, with mean gap $\Lambda$5. By NPI, however, G_HOPE and G_SDNE are best overall, with mean NPI values $\Lambda$6 and $\Lambda$7, respectively, followed by G_SPEC at $\Lambda$8. At $\Lambda$9 in the NPI performance profile, G_SDNE is best on $\Pi_0=\{\pi_0^1,\ldots,\pi_0^\Lambda\}$0 of instances, G_HOPE on $\Pi_0=\{\pi_0^1,\ldots,\pi_0^\Lambda\}$1, G_SPEC on $\Pi_0=\{\pi_0^1,\ldots,\pi_0^\Lambda\}$2, and G_N2V on $\Pi_0=\{\pi_0^1,\ldots,\pi_0^\Lambda\}$3. The stochastic variants are markedly slower and substantially worse by NPI because recomputing embeddings each iteration is expensive (Charytitsch et al., 21 Aug 2025).

The second experiment compares G_HOPE and G_SDNE against GRASP2 and GRASP3.

Method	Mean gap / time	Mean NPI
GRASP2	$\Pi_0=\{\pi_0^1,\ldots,\pi_0^\Lambda\}$4 / $\Pi_0=\{\pi_0^1,\ldots,\pi_0^\Lambda\}$5 s	$\Pi_0=\{\pi_0^1,\ldots,\pi_0^\Lambda\}$6
GRASP3	$\Pi_0=\{\pi_0^1,\ldots,\pi_0^\Lambda\}$7 / $\Pi_0=\{\pi_0^1,\ldots,\pi_0^\Lambda\}$8 s	$\Pi_0=\{\pi_0^1,\ldots,\pi_0^\Lambda\}$9
G_HOPE	$$\pi_0^\lambda:V^\lambda\mapsto\{1,\ldots,|V^\lambda|\}$$0 / $$\pi_0^\lambda:V^\lambda\mapsto\{1,\ldots,|V^\lambda|\}$$1 s	$$\pi_0^\lambda:V^\lambda\mapsto\{1,\ldots,|V^\lambda|\}$$2
G_SDNE	$$\pi_0^\lambda:V^\lambda\mapsto\{1,\ldots,|V^\lambda|\}$$3 / $$\pi_0^\lambda:V^\lambda\mapsto\{1,\ldots,|V^\lambda|\}$$4 s	$$\pi_0^\lambda:V^\lambda\mapsto\{1,\ldots,|V^\lambda|\}$$5

These results indicate that GRASP3 attains the best average final gap, but GL-GRASP variants are much better by primal integral. The class-wise comparison is sharper: GRASP3 dominates in lower-layer or easier cases, especially with $$\pi_0^\lambda:V^\lambda\mapsto\{1,\ldots,|V^\lambda|\}$$6 or $$\pi_0^\lambda:V^\lambda\mapsto\{1,\ldots,|V^\lambda|\}$$7 layers and low density $$\pi_0^\lambda:V^\lambda\mapsto\{1,\ldots,|V^\lambda|\}$$8, whereas G_HOPE and G_SDNE surpass GRASP3 on average gap for larger and denser instances, such as $$\pi_0^\lambda:V^\lambda\mapsto\{1,\ldots,|V^\lambda|\}$$9 and $\mathcal{D}=(G,\Pi_0)$00. The Wilcoxon signed-rank tests at $\mathcal{D}=(G,\Pi_0)$01 significance show no significant difference in gap among GRASP3, G_HOPE, and G_SDNE, but GRASP3 is significantly slower than G_HOPE and G_SDNE (Charytitsch et al., 21 Aug 2025).

The third experiment strengthens the scalability claim. On the denser large instances, G_HOPE and G_SDNE both attain mean gap $\mathcal{D}=(G,\Pi_0)$02, compared with $\mathcal{D}=(G,\Pi_0)$03 for GRASP2, $\mathcal{D}=(G,\Pi_0)$04 for GRASP3, and $\mathcal{D}=(G,\Pi_0)$05 for Gurobi under the stated time limit. The paper reports that each of G_HOPE and G_SDNE achieved the best gap in $\mathcal{D}=(G,\Pi_0)$06 of the $\mathcal{D}=(G,\Pi_0)$07 instances. This suggests that the hybridization is particularly effective when density and structural complexity increase (Charytitsch et al., 21 Aug 2025).

An important nuance is that the learned construction is not superior as a pure constructor. The average construction-only gaps are $\mathcal{D}=(G,\Pi_0)$08 for $\mathcal{D}=(G,\Pi_0)$09, $\mathcal{D}=(G,\Pi_0)$10 for $\mathcal{D}=(G,\Pi_0)$11, $\mathcal{D}=(G,\Pi_0)$12 for $\mathcal{D}=(G,\Pi_0)$13, and $\mathcal{D}=(G,\Pi_0)$14 for $\mathcal{D}=(G,\Pi_0)$15. After local search, however, GRASP3, G_HOPE, and G_SDNE end with very similar average crossing counts, while the learning-based variants are substantially faster. The main empirical contribution of graph representation learning is therefore a more favorable quality–time balance, not a better initial crossing count in isolation (Charytitsch et al., 21 Aug 2025).

4. Complexity landscape and closely related formal variants

A broader understanding of C-IGDP is shaped by adjacent constrained extension problems in straight-line, orthogonal, grid, and region-constrained drawing. One especially relevant model is Graph in Polygon: given a planar graph $\mathcal{D}=(G,\Pi_0)$16 and a polygonal region $\mathcal{D}=(G,\Pi_0)$17, with some vertices fixed on the boundary of $\mathcal{D}=(G,\Pi_0)$18, determine whether the remaining vertices can be placed to obtain a planar straight-line drawing inside $\mathcal{D}=(G,\Pi_0)$19. This model generalizes partial planar drawing extension and is directly described as a geometric feasibility version of constrained incremental planar drawing with fixed vertex placements, region constraints, straight-line edges, and exact planarity preservation. Its main result is that Graph in Polygon is $\mathcal{D}=(G,\Pi_0)$20-complete, whether the graph is given abstractly or with a combinatorial embedding, and the paper also proves that integer-coordinate inputs may force irrational coordinates in every valid drawing. The complexity remains open for a simply connected region (Lubiw et al., 2018).

A second nearby model is Bend-Minimal Orthogonal Extension (BMOE), where a connected subgraph $\mathcal{D}=(G,\Pi_0)$21 is already drawn orthogonally and planarly, and the task is to insert the missing vertices and edges so that the full graph has a planar orthogonal drawing extending $\mathcal{D}=(G,\Pi_0)$22, while minimizing bends on the inserted part. The paper establishes fixed-parameter tractability with respect to

$\mathcal{D}=(G,\Pi_0)$23

via a reduction to a single-face insertion problem, bend-equivalent sectors, a sector graph of bounded treewidth, a global point-grid discretization, and dynamic programming. Its final result states that BMOE can be solved in time $\mathcal{D}=(G,\Pi_0)$24, where $\mathcal{D}=(G,\Pi_0)$25 is the number of feature points of $\mathcal{D}=(G,\Pi_0)$26. This is a direct positive result for a specialized but important constrained incremental model (Bhore et al., 2023).

Topologically-Safe Snapping (TSS) studies a different re-optimization regime: a plane straight-line drawing must be moved to integer grid points inside a bounding box while preserving topological equivalence, straight-line edges, and cyclic orders around vertices, with objective

$\mathcal{D}=(G,\Pi_0)$27

The paper proves NP-hardness for total Manhattan movement, total Euclidean movement, and Euclidean maximum movement, and provides an ILP whose constraints enforce vertex distinctness, noncrossing of nonincident edges, direction assignments, and preservation of cyclic order. In a C-IGDP interpretation, this is an embedding-preserving incremental redraw problem rather than an insertion problem (Löffler et al., 2016).

Region-constrained clustered drawing yields yet another complexity boundary. For a flat clustered graph $\mathcal{D}=(G,\Pi_0)$28 and a disk arrangement $\mathcal{D}=(G,\Pi_0)$29, every simple clustered graph with a $\mathcal{D}=(G,\Pi_0)$30-framed embedding has a $\mathcal{D}=(G,\Pi_0)$31-framed straight-line drawing homeomorphic to that embedding when the disk arrangement has no pipe-disk intersections. If that condition is dropped, deciding existence becomes NP-hard, even when the clustered graph has maximum degree $\mathcal{D}=(G,\Pi_0)$32 in the disk setting. This result is especially relevant when incremental constraints are interpreted as progressively imposed territorial regions and routing corridors (Mchedlidze et al., 2018).

Taken together, these results suggest that the computational status of “constrained incremental drawing” depends strongly on the representation model. Exact straight-line extension with region constraints can reach $\mathcal{D}=(G,\Pi_0)$33-completeness, orthogonal extension with a small missing subgraph can be fixed-parameter tractable, embedding-preserving grid snapping is NP-hard, and clustered region assignments exhibit a sharp positive/negative boundary depending on whether pipes intersect foreign disks.

5. Dynamic, interactive, and continuous optimization perspectives

Several papers do not formalize C-IGDP under that exact name, but they articulate mechanisms that are directly relevant to broader constrained incremental drawing. One line uses force-directed optimization with soft and hard constraints. A constraint-based framework for grid-like layout combines a P-stress objective with additional soft terms,

$\mathcal{D}=(G,\Pi_0)$34

and hard constraints handled by projection. Its Adaptive Constrained Alignment (ACA) mechanism greedily inserts hard separated alignments such as

$\mathcal{D}=(G,\Pi_0)$35

with conflict resolution that distinguishes definite user constraints from tentative automatically generated constraints. The system is implemented in Dunnart, where automatic layout runs continuously in a background worker thread, allowing the layout to adapt immediately to user-specified changes to positions or constraints. This is a practically incremental, editor-oriented model rather than a formal dynamic algorithm (Kieffer et al., 2013).

A complementary optimization engine is pairwise stochastic gradient descent for stress minimization. The stress objective is

$\mathcal{D}=(G,\Pi_0)$36

and the per-pair clipped update is

$\mathcal{D}=(G,\Pi_0)$37

where $\mathcal{D}=(G,\Pi_0)$38. The paper argues that the pairwise decomposition makes constrained layout easier than majorization because new terms, emphasis weights, or local corrections can be added without redesigning a global solver. It also states that the method is well suited to real-time applications such as interactive layout and to dynamical situations where the structure of the graph changes with time. This suggests a reusable optimization core for C-IGDP-style systems, especially when warm starts or selective updates are desirable (Zheng et al., 2017).

A more explicitly online model appears in streamed graph drawing. Here a planar graph is revealed as an edge stream; the drawing must be updated after each arrival using straight-line segments with endpoints at integer grid points. The paper shows that if no vertex moves are allowed, even trees may require $\mathcal{D}=(G,\Pi_0)$39 area, whereas allowing a small amount of recourse yields polynomial-area streamed drawings for trees, tree-maps, and outerplanar graphs. For trees, the paper gives several tradeoffs, including $\mathcal{D}=(G,\Pi_0)$40 area with $\mathcal{D}=(G,\Pi_0)$41 vertex moves per insertion if bulk moves of size $\mathcal{D}=(G,\Pi_0)$42 are allowed, and polynomial area with $\mathcal{D}=(G,\Pi_0)$43 vertex moves per insertion if bulk moves are not allowed. The key reduction is to the file maintenance problem, where coordinate changes are treated as relabelings in an ordered set (Goodrich et al., 2013).

These three lines of work point to distinct incremental paradigms. One treats constraints as projection and revocable hard alignments, one treats them as modular objective terms updated by SGD, and one treats geometric stability as bounded recourse in an online stream. This suggests that “incrementality” in graph drawing is not a single notion: it may mean preserving exact geometry, limiting the number of moved coordinates, maintaining a fixed subdrawing, or continuously adapting a constrained optimization state.

6. Scope, interpretation, and open directions

A recurring misconception is to treat C-IGDP as a single universally agreed model. The literature instead supports two levels of meaning. In the narrow sense, C-IGDP is the multilayer crossing-minimization problem with relative-order preservation and displacement bound $\mathcal{D}=(G,\Pi_0)$44 on original vertices (Charytitsch et al., 21 Aug 2025). In the broader sense, the term functions as a useful umbrella for constrained extension, redraw, and update problems in which an existing drawing or embedding restricts feasible insertions or adjustments. This broader use is interpretive rather than terminologically uniform across papers.

A second misconception is to equate all “incremental” drawing with online edge-by-edge updates. The surveyed work spans several non-equivalent regimes: batch insertion of a missing subgraph into a fixed orthogonal drawing (Bhore et al., 2023), embedding-preserving coordinate snapping of an already drawn plane graph (Löffler et al., 2016), continuous editor-driven reoptimization under soft and hard constraints (Kieffer et al., 2013), and streamed insertion with bounded recourse (Goodrich et al., 2013). The canonical C-IGDP formulation is itself a batch optimization problem on multilayer drawings rather than a fully dynamic data structure problem (Charytitsch et al., 21 Aug 2025).

A third misconception is that exact constrained extension is merely NP-hard in an ordinary discrete sense. The Graph in Polygon result shows a stronger phenomenon: exact straight-line extension in a polygonal region is $\mathcal{D}=(G,\Pi_0)$45-complete, and irrational coordinates may be unavoidable even for integer input. This indicates that some constrained extension tasks are fundamentally real-algebraic rather than purely combinatorial (Lubiw et al., 2018).

Current research directions are correspondingly heterogeneous. Within the hierarchical C-IGDP model, the 2025 hybrid GRL–GRASP work does not modify the exact formulation or local-search neighborhoods; it contributes a new construction guidance mechanism and shows the strongest gains on denser and larger instances. The paper also notes that embedding objectives are not tailored to crossing minimization and that embeddings are static rather than adapted to search state, which suggests further room for specialized learning strategies (Charytitsch et al., 21 Aug 2025). In related extension models, open questions include the complexity of Graph in Polygon for a simply connected region, the complexity of the tree case, and the status of open-region partial drawing extension (Lubiw et al., 2018). In orthogonal extension, the established FPT result is constructive but remains specialized to planar orthogonal drawings of maximum degree $\mathcal{D}=(G,\Pi_0)$46, fixed partial geometry, and batch insertion of a small missing subgraph (Bhore et al., 2023).

The present state of the subject therefore supports a layered view. At its core, C-IGDP is a constrained multilayer crossing-minimization problem motivated by mental-map preservation. Around that core lies a technically rich neighboring literature on partial drawing extension, constrained reoptimization, streamed updates, orthogonal insertion, region-constrained realization, and hybrid optimization. The combined picture is that constrained incrementality in graph drawing is not governed by a single algorithmic principle; rather, its behavior depends on which invariants are frozen, which movements are permitted, and whether the representation is straight-line, orthogonal, clustered, grid-based, or streaming.