Morphing Graph Drawings in the Presence of Point Obstacles

TL;DR

This work introduces and analyzes obstacle-avoiding planar graph morphs, where a fixed set of point obstacles must be avoided during a continuous, crossing-free deformation between two drawings of the same plane graph. It establishes that deciding the existence of such a morph is NP-hard, even when only three vertices move, by a gadget-based reduction from 3-SAT, and broadens the understanding of when obstacle configurations cannot block morphs, including non-blockable forests and canonical-cycle constructions. Key contributions include tight lower bounds on the number of obstacles needed to block $3$-cycles, a detailed study of shifted cycle drawings with and without free vertices, and a comprehensive NP-hardness framework using variable, literal, clause, split, crossing, and synchronization gadgets. The results illuminate the computational complexity introduced by obstacles in morphing tasks and lay out several open problems regarding obstacle counts, graph classes, and the nature of morphs, with implications for 2D–3D–2D morph pipelines and related geometric deformation workflows.

Abstract

A crossing-free morph is a continuous deformation between two graph drawings that preserves straight-line pairwise noncrossing edges. Motivated by applications in 3D morphing problems, we initiate the study of morphing graph drawings in the plane in the presence of stationary point obstacles, which need to be avoided throughout the deformation. As our main result, we prove that it is NP-hard to decide whether such an obstacle-avoiding 2D morph between two given drawings of the same graph exists. In fact, this statement remains true even in the severely restricted special case where only three vertices have to change positions. This is in sharp contrast to the classical case without obstacles, where there is an efficiently verifiable (necessary and sufficient) criterion for the existence of a morph. Further, we provide several combinatorial results related to conditions under which the existence of a morph between two drawings of a graph can or cannot be prevented by the placement of a given number of point obstacles.

Figures (17)

• Figure 1: Two distinct drawings $\Gamma_1$ and $\Gamma_2$ of a plane cycle $(a,b,c)$ and a set $P$ consisting of three internal obstacles (blue crosses) and three external obstacles (red stars), which are compatible with $\Gamma_1$ and $\Gamma_2$. There exists no planar morph between $\Gamma_1$ and $\Gamma_2$ that avoids $P$, i.e., the drawings are blocked by $P$. (For a formal justification, see \ref{['prop:block-label-shift-C3-2']}.)
• Figure 2: (a) Two simple closed curves $\Gamma_1$ and $\Gamma_2$ both containing the two obstacles marked by blue crosses and neither containing the obstacle marked by a red star. The curves $\Gamma_1$ and $\Gamma_2$ cannot be continuously deformed into each other without passing over one of the obstacles and while preserving simplicity. (The corresponding continuous deformation of the geodesic joining the two blue internal obstacles within the closed curve would transform the curve $g_1$ into $g_2$ while keeping the endpoints fixed and without passing over the red external obstacle, which is impossible.) Consequently, the two drawings of a 4-cycle in (b) are blocked by the set of three obstacles; this set is not compatible with the two drawings.
• Figure 3: Two drawings of $C_8$. If the (red) shaded regions are densely filled with obstacles, the drawing on the left is essentially locked in place---it cannot be morphed planarly to a substantially different drawing without intersecting the obstacle regions. In particular, it cannot be morphed to the drawing on the right, even though the right drawing contains two free vertices (and the obstacles are compatible with the two drawings).
• Figure 4: Construction of the canonical triangle.
• Figure 5: A morph of a canonical drawing of a 3-cycle with vertices $u$, $v$, $w$ placed at $c$, $a$, $b$, respectively, to a shifted version where the vertices are placed at $a$, $b$, $c$, respectively. In the subfigures, we used the notation $x/y$ to mean that vertex $y$ is placed at point $x$.

Theorems & Definitions (20)

• Theorem 1
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• Proposition 6
• proof
• Proposition 7