[Paper Review] Contact Representations of Sparse Planar Graphs

Md. Jawaherul Alam, David Eppstein+5 more|arXiv (Cornell University)|Jan 1, 2015

Structural Analysis and Optimization29 references6 citations

TL;DR

This paper investigates contact representations of sparse planar graphs using circular arcs (CCA-representations), where vertices are represented by non-overlapping circular arcs and edges by interior-point contacts. It proves that all plane (2,2)-sparse graphs and certain (2,1)-tight, (2,0)-tight, and duals of (2,3)- and (2,4)-tight graphs admit CCA-representations, while recognizing that not all (2,0)-tight multigraphs do. It further shows that every plane (2,0)-sparse graph has a wedge-based contact representation, and identifies NP-completeness for right-angled wedge representations under degree constraints.

ABSTRACT

We study representations of graphs by contacts of circular arcs, CCA-representations for short, where the vertices are interior-disjoint circular arcs in the plane and each edge is realized by an endpoint of one arc touching the interior of another. A graph is (2,k)-sparse if every s-vertex subgraph has at most 2s - k edges, and (2, k)-tight if in addition it has exactly 2n - k edges, where n is the number of vertices. Every graph with a CCA- representation is planar and (2, 0)-sparse, and it follows from known results on contacts of line segments that for k >= 3 every (2, k)-sparse graph has a CCA-representation. Hence the question of CCA-representability is open for (2, k)-sparse graphs with 0 <= k <= 2. We partially answer this question by computing CCA-representations for several subclasses of planar (2,0)-sparse graphs. In particular, we show that every plane (2, 2)-sparse graph has a CCA-representation, and that any plane (2, 1)-tight graph or (2, 0)-tight graph dual to a (2, 3)-tight graph or (2, 4)-tight graph has a CCA-representation. Next, we study CCA-representations in which each arc has an empty convex hull. We characterize the plane graphs that have such a representation, based on the existence of a special orientation of the graph edges. Using this characterization, we show that every plane graph of maximum degree 4 has such a representation, but that finding such a representation for a plane (2, 0)-tight graph with maximum degree 5 is an NP-complete problem. Finally, we describe a simple algorithm for representing plane (2, 0)-sparse graphs with wedges, where each vertex is represented with a sequence of two circular arcs (straight-line segments).

Motivation & Objective

• To determine which classes of sparse planar graphs admit contact representations using non-overlapping circular arcs (CCA-representations).
• To resolve the open question of whether every (2,0)-sparse planar graph has a CCA-representation, particularly when allowing for graph embedding changes.
• To investigate the existence of contact representations using wedges (polyline segments with one bend), especially with geometric constraints like right angles.
• To characterize the conditions under which a plane graph admits a CCA-representation with arcs having empty convex hulls.
• To examine the computational complexity of constructing CCA-representations with specific geometric properties, such as right-angled wedges.

Proposed method

• Uses graph-theoretic sparsity conditions: (p,k)-sparse and (p,k)-tight graphs, focusing on (2,k)-sparsity for k ∈ {0,1,2,3,4}.
• Applies structural lemmas on augmentations and duality of (2,k)-tight graphs to identify representable subclasses.
• Employs 2−-orientations of (2,0)-sparse graphs to construct wedge-based contact representations via straight-line drawings.
• Introduces a characterization based on edge orientations to determine when a CCA-representation exists with empty convex hulls for the arcs.
• Reduces the problem of right-angled wedge representations to geometric feasibility, proving NP-completeness for graphs with maximum degree 5.
• Uses geometric arguments involving pseudo-triangles and angle sums to show infeasibility of right-angled wedge representations for certain (2,0)-tight graphs like the octahedron.

Experimental results

Research questions

• RQ1 Do all (2,0)-sparse planar graphs admit a CCA-representation, even when the embedding is not fixed?
• RQ2 Which subclasses of (2,0)-sparse planar graphs guarantee a CCA-representation, and what structural properties enable this?
• RQ3 Can every plane (2,0)-sparse graph be represented using wedges (polyline segments with one bend), and what constraints affect such representations?
• RQ4 What is the computational complexity of determining whether a (2,0)-tight planar graph admits a CCA-representation with right-angled wedges?
• RQ5 Under what conditions does a CCA-representation exist where each arc has an empty convex hull?

Key findings

• Every plane (2,2)-sparse graph admits a CCA-representation.
• Any (2,1)-tight or (2,0)-tight plane graph whose dual is (2,3)- or (2,4)-tight also admits a CCA-representation.
• A (2,0)-tight plane multigraph with maximum degree 5 exists that does not admit a CCA-representation, showing that not all (2,0)-tight graphs are CCA-representable.
• Every plane (2,0)-sparse graph has a contact representation using wedges, where each vertex is represented by a two-segment polyline.
• Finding a CCA-representation with right-angled wedges for a (2,0)-tight graph with maximum degree 5 is NP-complete.
• The octahedron graph, a (2,0)-tight graph with triangular faces, cannot have a contact representation with right-angled wedges due to geometric constraints on outer boundary angles.