arXiv subjects

Giordano Andreola

Publications and source records attributed to Giordano Andreola.

2 recordsLinked to original sources

arXiv2026

Extending Biconnected Straight-Line Planar Drawings

The Partial Drawing Extensibility problem, for short PDE, takes as input a triple $\langle G,H,Γ_H\rangle$, where $G$ is a planar graph, $H$ is a subgraph of $G$, and $Γ_H$ is a straight-line planar drawing of $H$, and asks whether $Γ_H$ can be extended to a straight-line planar drawing of $G$. Patrignani [Int. J. Found. Comput. Sci. (2006)] proved that the PDE problem is NP-hard, exploiting instances in which $H$ is highly disconnected. In this paper, we study the PDE problem under the requirement that the initial partial drawing $Γ_H$ is biconnected. We show that PDE remains NP-hard even for instances in which $H$ is a biconnected graph with faces of bounded size, $G$ is subcubic, and the part of $G$ that is not in $H$ consists of length-$2$ paths. The complexity of PDE remains however open when $H$ is connected (or even biconnected) if $G$ has a fixed embedding. In this setting both a polynomial-time algorithm or an NP-hardness proof seem to be elusive targets. As a step towards tackling this problem, we study instances of PDE in which $H$ is biconnected, $G$ has a fixed embedding, and the rest of the graph consists of $p$ length-2 paths, and present an $O(p^2 n)$-time algorithm, a result in sharp contrast with the NP-hardness of the variable embedding setting. Moreover, with an approach based on the Existential Theory of the Reals, we show that, if $H$ is biconnected, the problem is FPT parameterized by the vertex cover number of $G$, both in a fixed and in a variable embedding setting.

cs.CG↗

arXiv2025

A Walk on the Wild Side: a Shape-First Methodology for Orthogonal Drawings

Several algorithms for the construction of orthogonal drawings of graphs, including those based on the Topology-Shape-Metrics (TSM) paradigm, tend to prioritize the minimization of crossings. This emphasis has two notable side effects: some edges are drawn with unnecessarily long sequences of segments and bends, and the overall drawing area may become excessively large. As a result, the produced drawings often lack geometric uniformity. Moreover, orthogonal crossings are known to have a limited impact on readability, suggesting that crossing minimization may not always be the optimal goal. In this paper, we introduce a methodology that 'subverts' the traditional TSM pipeline by focusing on minimizing bends. Given a graph $G$, we ideally seek to construct a rectilinear drawing of $G$, that is, an orthogonal drawing with no bends. When not possible, we incrementally subdivide the edges of $G$ by introducing dummy vertices that will (possibly) correspond to bends in the final drawing. This process continues until a rectilinear drawing of a subdivision of the graph is found, after which the final coordinates are computed. We tackle the (NP-complete) rectilinear drawability problem by encoding it as a SAT formula and solving it with state-of-the-art SAT solvers. If the SAT formula is unsatisfiable, we use the solver's proof to determine which edge to subdivide. Our implementation, DOMUS, which is fairly simple, is evaluated through extensive experiments on small- to medium-sized graphs. The results show that it consistently outperforms OGDF's TSM-based approach across most standard graph drawing metrics.

cs.CG↗