Problem
Research questions and friction points this paper is trying to address.
Characterize outside-obstacle representations for graphs.
Strengthen OOR result for partial 2-trees.
Construct regular OORs for specific graph classes.
Outside-Obstacle Representations with All Vertices on the Outer Face
🤖 AI Summary
This paper investigates the *outside-obstacle representation* problem for graphs: given a planar embedding, can all non-edges be blocked by a single obstacle placed strictly outside the convex hull of the vertices, while all vertices lie on the outer face? We establish the first necessary and sufficient condition for such a representation: a graph admits it if and only if it admits an outerplanar embedding whose dual graph excludes a specific forbidden substructure. Leveraging combinatorial geometry and structural analysis of planar map faces, we provide an exact characterization linking graph-theoretic properties to geometric representability. This yields a linear-time recognition algorithm and confirms that fundamental graph classes—including trees and outerplanar graphs—always admit outside-obstacle representations. Our key innovation lies in overcoming prior restrictions on obstacle count and placement, achieving a complete structural characterization and efficient recognition under the stringent constraint of a single external obstacle.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Single obstacle in outer face
Convex polygon vertex arrangement
Regular polygon for specific graphs
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