Grid-Obstacle Representations with Connections to Staircase Guarding

TL;DR

This paper investigates grid-obstacle graph representations, demonstrating smaller grid sizes for planar and general graphs, exploring bipartite cases, and establishing NP-hardness of staircase guarding in 2D.

Contribution

It introduces more efficient grid-obstacle representations for planar and general graphs and connects these to staircase guarding complexity.

Findings

Smaller grid sizes for 2D and 3D representations.

Existence of grid-obstacle representations for bipartite graphs without blocking.

NP-hardness of staircase guarding in 2D.

Abstract

In this paper, we study grid-obstacle representations of graphs where we assign grid-points to vertices and define obstacles such that an edge exists if and only if an $xy$-monotone grid path connects the two endpoints without hitting an obstacle or another vertex. It was previously argued that all planar graphs have a grid-obstacle representation in 2D, and all graphs have a grid-obstacle representation in 3D. In this paper, we show that such constructions are possible with significantly smaller grid-size than previously achieved. Then we study the variant where vertices are not blocking, and show that then grid-obstacle representations exist for bipartite graphs. The latter has applications in so-called staircase guarding of orthogonal polygons; using our grid-obstacle representations, we show that staircase guarding is \textsc{NP}-hard in 2D.