Circular-arc graphs and the Helly property

TL;DR

This work studies Helly properties in circular-arc graphs by leveraging conformal/PQM-tree representations that capture all normalized models. It provides an alternative Lin–Szwarcfiter proof and a fine-grained clique-type classification with a polynomial-time tester, then delivers an FPT algorithm for Helly Cliques with a kernel of size O(k^6) and ETH-based lower bounds, together with NP-completeness results. The approach hinges on the PQM-tree data structure, modular decompositions of G_{ov}, and geometric tools like the Trapezoid Lemma to connect combinatorial structure with model realizations. Overall, the results advance understanding of Helly-type properties in geometric intersection graphs and offer practical, scalable algorithms for Helly-related decision problems in circular-arc graphs.

Abstract

In this paper we investigate some problems related to the Helly properties of circular-arc graphs, which are defined as intersection graphs of arcs of a fixed circle. As such, circular-arc graphs are among the simplest classes of intersection graphs whose models might not satisfy the Helly property. In particular, some cliques of a circular-arc graph might be Helly in some but not all arc intersection models of the graph. Our first result is an alternative proof of a theorem by Lin and Szwarcfiter which asserts that for every circular-arc graph $G$ either every normalized model of $G$ satisfies the Helly property or no normalized model of $G$ satisfies this property. Further, we study the Helly properties of a single clique of a circular-arc graph $G$. We divide the cliques of $G$ into three types: a clique $C$ of $G$ is always-Helly/always-non-Helly/ambiguous if $C$ is Helly in every/no/(some but not all) normalized model of $G$. We provide a combinatorial description for the cliques of each type, and based on it, we devise a polynomial time algorithm which determines the type of a given clique. Finally, we study the Helly Cliques problem, in which we are given an $n$-vertex circular-arc graph $G$ and some of its cliques $C_1, \ldots, C_k$ and we ask if there is an arc intersection model of $G$ in which all the cliques $C_1, \ldots, C_k$ satisfy the Helly property. We show that: (1) the Helly Cliques problem admits a $2^{O(k\log{k})}n^{O(1)}$-time algorithm (that is, it is FPT when parametrized by the number of cliques given in the input), (2) assuming Exponential Time Hypothesis (ETH), the Helly Cliques problem cannot be solved in time $2^{o(k)}n^{O(1)}$, (3) the Helly Cliques problem admits a polynomial kernel of size $O(k^6)$. All our results use a data structure, called a PQM-tree, which maintains all normalized models of a circular-arc graph $G$.

Figures (28)

• Figure 2.1: From left to right: $\psi(v)$ and $\psi(u)$ are disjoint, $\psi(v)$ contains $\psi(u)$, $\psi(v)$ is contained in $\psi(u)$, $\psi(v)$ and $\psi(u)$ cover the circle, and $\psi(v)$ and $\psi(u)$ overlap.
• Figure 2.2: Mutual relation between the arcs $\psi(v)$ and $\psi(u)$ and between the corresponding oriented chords $\phi(v)$ and $\phi(u)$ for the cases: $v$ and $u$ are disjoint, $v$ contains $u$, $v$ is contained in $u$, $v$ and $u$ cover the circle, and $v$ and $u$ overlap, respectively.
• Figure 2.3: A collection $B$ of oriented chords and points and its reflection $B^R$. The set $B$ is represented by the circular word $cw(B) = s^0_4s^1_3s^0_2s^1_4s^0_1qs^1_2s^0_3ps^1_1$ and the set $B^R$ is represented by $cw^R(B) = s^0_1ps^1_3s^0_2qs^1_1s^0_4s^1_2s^0_3s^1_4$.
• Figure 2.4: A permutation model $(\tau^0,\tau^1) = (abc,acb)$ of the permutation graph $(\{a,b,c\}, \{a \sim b, a \sim c)\}$.
• Figure 3.1: A conformal model $\phi$ of $G$ and the circular order of the slots $\pi(\phi)$ in $\phi$.

Theorems & Definitions (62)

• Theorem 1.1: LinSchw06
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Theorem 2.2: DM41
• Theorem 2.3: Gal67
• Theorem 2.4: Gal67
• Theorem 2.5: Gal67
• Definition 3.1