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Outside-Obstacle Representations with All Vertices on the Outer Face

Oksana Firman, Philipp Kindermann, Jonathan Klawitter, Boris Klemz, Felix Klesen, Alexander Wolff

TL;DR

The paper investigates outside-obstacle representations (OORs) with a single outer obstacle, proving that every (partial) $2$-tree has a reducible $OOR$ whose vertices lie on the outer face and extending this to graphs of treewidth at most $2$. It introduces convex $OOR$ criteria, including the Consec utive-Neighbors Property (a sufficient condition) and a necessary Gap Condition, and uses these to analyze which graphs admit convex $OOR$s. A key result is the complete characterization that the complement of a tree has a convex $OOR$ if and only if the tree is a caterpillar, with the corresponding convex (indeed regular) representations; non-caterpillars do not admit convex $OOR$s. The authors also construct regular $OOR$s for various graph families (outerpaths, grids, and cacti), highlighting both potential and limits of convex and regular outer representations, and discuss open questions related to treewidth and broader graph classes.

Abstract

An obstacle representation of a graph $G$ consists of a set of polygonal obstacles and a drawing of $G$ as a visibility graph with respect to the obstacles: vertices are mapped to points and edges to straight-line segments such that each edge avoids all obstacles whereas each non-edge intersects at least one obstacle. Obstacle representations have been investigated quite intensely over the last few years. Here we focus on outside-obstacle representations (OORs) that use only one obstacle in the outer face of the drawing. It is known that every outerplanar graph admits such a representation. We strengthen this result by showing that every (partial) 2-tree has an OOR. We also consider restricted versions of OORs where the vertices of the graph form a convex polygon or even a regular polygon. We characterize when the complement of a tree and when a complete graph minus a simple cycle admits a convex OOR. We construct regular OORs for all (partial) outerpaths, cactus graphs, and grids.

Outside-Obstacle Representations with All Vertices on the Outer Face

TL;DR

The paper investigates outside-obstacle representations (OORs) with a single outer obstacle, proving that every (partial) -tree has a reducible whose vertices lie on the outer face and extending this to graphs of treewidth at most . It introduces convex criteria, including the Consec utive-Neighbors Property (a sufficient condition) and a necessary Gap Condition, and uses these to analyze which graphs admit convex s. A key result is the complete characterization that the complement of a tree has a convex if and only if the tree is a caterpillar, with the corresponding convex (indeed regular) representations; non-caterpillars do not admit convex s. The authors also construct regular s for various graph families (outerpaths, grids, and cacti), highlighting both potential and limits of convex and regular outer representations, and discuss open questions related to treewidth and broader graph classes.

Abstract

An obstacle representation of a graph consists of a set of polygonal obstacles and a drawing of as a visibility graph with respect to the obstacles: vertices are mapped to points and edges to straight-line segments such that each edge avoids all obstacles whereas each non-edge intersects at least one obstacle. Obstacle representations have been investigated quite intensely over the last few years. Here we focus on outside-obstacle representations (OORs) that use only one obstacle in the outer face of the drawing. It is known that every outerplanar graph admits such a representation. We strengthen this result by showing that every (partial) 2-tree has an OOR. We also consider restricted versions of OORs where the vertices of the graph form a convex polygon or even a regular polygon. We characterize when the complement of a tree and when a complete graph minus a simple cycle admits a convex OOR. We construct regular OORs for all (partial) outerpaths, cactus graphs, and grids.
Paper Structure (6 sections, 4 theorems, 1 equation, 5 figures)

This paper contains 6 sections, 4 theorems, 1 equation, 5 figures.

Table of Contents

  1. Introduction
  2. Previous Work.
  3. Contribution.
  4. Notation.
  5. Outside-Obstacle Representations for Partial 2-Trees
  6. Convex Outside Obstacle Representations

Key Result

Theorem 2

Every $2$-tree admits a reducible OOR with all vertices on the outer face.

Figures (5)

  • Figure 1: Inside- and outside-obstacle representations of graphs. The gray regions represent the obstacles; the dashed line segments represent the non-edges.
  • Figure 2: Non-convex, circular, and regular outside-obstacle representations (OORs). Graph edges are solid black, non-edges are dashed red (in (b) only one non-edge is highlighted).
  • Figure 3: The so-called gyroelongated square bipyramid $X_4$, which has treewidth 4, does not admit a representation with a single obstacle.
  • Figure 5: Examples for (\ref{['fig:conditions:cnp']}) the consecutive-neighbors property and (\ref{['fig:conditions:candidategap']}) a candidate gap.
  • Figure 6: The Petersen graph does not admit a convex OOR.

Theorems & Definitions (4)

  • Theorem 2
  • Lemma 3: Consecutive-neighbors property
  • Lemma 4: Gap condition
  • Theorem 5