Recognition of Unit Segment and Polyline Graphs is $\exists\mathbb{R}$-Complete

TL;DR

This work proves that recognizing intersection graphs of unit segments and of polylines with a fixed number of bends is ER-complete, placing these natural geometric classes at the same level of difficulty as other ER-complete recognition problems. The authors develop cycle-based tools (cycle representations, boundary traces) and a reduction from pseudoline stretchability to build graphs whose realizability by unit segments or k-bend polylines corresponds to the stretchability of a given arrangement. ER-membership is established via both explicit ETR encodings and real-witness verifications, while ER-hardness is shown through careful constructions (probes, connectors, canvases) that enforce global order constraints on realizations. The results refine the ER/NP landscape for geometric graph classes, discuss implications for coordinate size and witness representations, and propose conjectures about the role of curvature and restricted graph classes in recognition complexity.

Abstract

Given a set of objects $O$ in the plane, the corresponding intersection graph is defined as follows. Each object defines a vertex and an edge joins two vertices whenever the corresponding objects intersect. We study here the case of unit segments and polylines with exactly $k$ bends. In the recognition problem, we are given a graph and want to decide whether the graph can be represented as an intersection graph of certain geometric objects. In previous work it was shown that various recognition problems are $\exists\mathbb{R}$-complete, leaving unit segments and polylines among the few remaining natural cases where the recognition complexity remained open. We show that recognition for both families of objects is $\exists\mathbb{R}$-complete.

Figures (16)

• Figure 1: A fictional illustration of mobile coverage of Switzerland using cell towers.
• Figure 2: Each box represents a different geometric intersection graph class. Upward edges between classes express containment. Those marked in green can be recognized in polynomial time. Those in blue are known to be ER-complete. The ones in gray are $\textsf{NP}$-complete, and the orange ones are the new results presented in this paper.
• Figure 3: The pseudoline arrangement on the left is combinatorially equivalent to the (truncated) line arrangement on the right; hence, it is stretchable.
• Figure 4: Top row: Representations $r(C)$ by unit segments, $1$-polylines and strings, respectively. Bottom row: The representation of the top row with two boundary curves added; one on the interior and one on the exterior of the representation. Left: The interior curve traces the Jordan arcs $abcdcdefg$. The exterior curve traces $ababcbcdedefefgfgag$. Middle: The interior curve traces the Jordan arcs $abcdcdef$. The exterior curve traces $ababcdcdefefaf$. Right: The interior curve traces the Jordan arcs $abcdcdcd$. The exterior curve traces $ababcbcdad$.
• Figure 5: A geometric realization of the cycle on 6 vertices using Jordan arcs.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2
• Conjecture A
• Conjecture B
• Lemma 3
• proof
• Definition 4
• Lemma 5
• proof
• Definition 6