Characterization of Circular-arc Graphs: III. Chordal Graphs

TL;DR

This work resolves a major open problem on characterizing chordal circular-arc graphs by identifying the complete set of minimal forbidden induced subgraphs. It develops a structural framework around McConnell flipping, introducing the $G^{s}$ and $G^{K}$ constructions to translate circular-arc constraints into interval-graph patterns. The authors define the ⊗ family, built from gadgets $D_k$ derived from $\overline{S_{k+1}}$ and connected by paths around a center, showing that $\otimes(a_{0},\ldots,a_{2p-1})$ with $(a_{i})\neq(1,1),(1,2)$ are minimal chordal forbidden subgraphs, while $\otimes(1,1)$ and $\otimes(1,2)$ are circular-arc graphs. Alongside the infinite family $\overline{S_k^{+}}$ ($k\ge3$) and a set of small graphs like the long claw and whipping top$^\star$, this yields a complete dichotomy: a chordal graph is circular-arc exactly when it avoids these forbidden configurations, thereby resolving the open problem and providing a principled recognition framework.

Abstract

We identify all minimal chordal graphs that are not circular-arc graphs, thereby resolving one of ``the main open problems'' concerning the structures of circular-arc graphs as posed by Dur{á}n, Grippo, and Safe in 2011. The problem had been attempted even earlier, and previous efforts have yielded partial results, particularly for claw-free graphs and graphs with an independence number of at most four. The answers turn out to have very simple structures: all the nontrivial ones belong to a single family. Our findings are based on a structural study of McConnell's flipping, which transforms circular-arc graphs into interval graphs with certain representation patterns.

Figures (28)

• Figure 1: A circular-arc graph and its two circular-arc models. In (b), any two arcs for vertices $\{2, 4, 6\}$ cover the circle; in (c), the three arcs for vertices $\{2, 4, 6\}$ do not share any common point.
• Figure 2: Minimal chordal graphs that are not interval graphs. A $\dag$ graph or a $\ddag$ graph contains at least six vertices.
• Figure 3: The complements of $k$-suns, for $k = 2, 3, 4, 5$.
• Figure 4: The gadget $D_{k}$. Lines among solid nodes are omitted for clarity.
• Figure 5: The graph $\otimes(a_{0}, a_{1}, \ldots, a_{2 p - 1})$. Inside each shadowed ellipse is a gadget. Each clique vertex (solid node) of a gadget is adjacent to all vertices not in the gadget.

Theorems & Definitions (52)

• Theorem 1.1: lekkerkerker-62-interval-graphs
• Proposition 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Definition : Collateral edge
• Definition : Witness
• Lemma 2.1
• proof