On the structure of normalized models of circular-arc graphs -- Hsu's approach revisited

TL;DR

A data-structure is devised, called PQM-tree, that maintains the set of all normalized models of a circular-arc graph and it is shown that the PQM-tree of a circular-arc graph can be computed in linear time.

Abstract

Circular-arc graphs are the intersection graphs of arcs of a circle. The main result of this work describes the structure of all \emph{normalized intersection models} of circular-arc graphs. Normalized models of a circular-arc graph reflect the neighborhood relation between its vertices and can be seen as its canonical representations; in particular, any intersection model can be made normalized by possibly extending some of its arcs. We~devise a data-structure, called \emph{PQM-tree}, that maintains the set of all normalized models of a circular-arc graph. We show that the PQM-tree of a circular-arc graph can be computed in linear time. Finally, basing on PQM-trees, we provide a linear-time algorithm for the canonization and the isomorphism problem for circular-arc graphs. We describe the structure of the normalized models of circular-arc graphs using an approach proposed by Hsu~[\emph{SIAM J. Comput. 24(3), 411--439, (1995)}]. In the aforementioned work, Hsu claimed the construction of decomposition trees representing the set of all normalized intersection models of circular-arc graphs and an $\mathcal{O}(nm)$ time isomorphism algorithm for this class of graphs. However, the counterexample given in~[\emph{Discrete Math. Theor. Comput. Sci., 15(1), 157--182, 2013}] shows that Hsu's isomorphism algorithm is incorrect. Also, in a companion paper we show that the decomposition trees proposed by Hsu are not constructed correctly; in particular, we showed that there are circular-arc graphs whose all normalized models do not follow the description given by Hsu.

Figures (50)

• Figure 2.1: To the left: a collection $B = \{a_1,a_2,a_3,c,p,q\}$ consisiting of three arcs $a_1,a_2,a_3$, a chord $c$, and two points $p, q$ is represented by the circular word $\tau(B) \equiv a_2^0pa_1^1ca^0_3a_2^1a^1_3qa_1^0c$. The word $\tau(B^R) \equiv ca_1^1qa^0_3a_2^0a^1_3ca_1^0pa_2^1$ is the reflection of $\tau(B)$. To the right: the same collection in which the arcs are replaced by the corresponding oriented chords.
• Figure 2.2: To the left: intersection model $\psi$ of a circular-arc graph $G = (V,E)$, where $V = \{v_1,\ldots,v_6\}$ and $E = \{ v_iv_{i+1}: i \in [5]\}\cup \{v_6v_1\}$. We have $\psi \Vert \{v^0_1,v^1_1, v^0_6,v^1_6\} \equiv v_6^0v_1^0v_6^1v^1_1$ (in red). The set $\{v^1_3,v^0_4,v^0_5\}$ is contiguous in $\psi$ (in blue) and we have $\psi|\{v^1_3,v^0_4,v^0_5\} = v^0_4v^1_3v^0_5$. To the right: the corresponding oriented chord model.
• Figure 2.3: Intersection model $(\tau^0,\tau^1) = (abc,acb)$ of the permutation graph $(\{a,b,c\}, \{a \sim b, a \sim c)\}$ corresponding to the transitive orientations $\{a \prec b, a \prec c\}$ and $\{b < c\}$ of $(V,{\sim})$ and $(V,{\parallel}\}$, respectively.
• Figure 2.4: 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 3.1: The transformation of the arc $\psi(v)$ into the oriented chord $\phi(v)$.

Theorems & Definitions (85)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1: Gal67
• Theorem 2.2: Gal67
• Theorem 2.3: Gal67
• Theorem 2.4: DM41
• Claim 2.6
• proof
• Definition 2.7
• Definition 3.1