Characterization of Circular-arc Graphs: II. McConnell Flipping

TL;DR

The paper analyzes McConnell's flipping transformation from circular-arc graphs to interval graphs to advance recognition and forbidden-subgraph characterization. It proves that a graph $G$ is circular-arc iff there exists a clique $K$ such that the transformed graph $G^K$ is an interval graph satisfying a double-extension property; in the $C_4$-free case, a weaker star condition suffices and a finite set of forbidden configurations precisely determines when the condition fails. The authors develop a constructive, PQ-tree based approach to certify interval representations that satisfy the star condition and show how these tools yield polynomial-time certifying recognition for $C_4$-free circular-arc graphs, with implications for chordal and Helly subclasses. Overall, the work connects structural patterns in flipping to a robust framework for recognizing circular-arc graphs and identifying minimal obstructions.

Abstract

McConnell [FOCS 2001] presented a flipping transformation from circular-arc graphs to interval graphs with certain patterns of representations. Beyond its algorithmic implications, this transformation is instrumental in identifying all minimal graphs that are not circular-arc graphs. We conduct a structural study of this transformation, and for $C_{4}$-free graphs, we achieve a complete characterization of these patterns. This characterization allows us, among other things, to identify all minimal chordal graphs that are not circular-arc graphs in a companion paper.

Figures (14)

• 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 graphs that are not interval graphs. A $\dag$ graph or a $\ddag$ graph contains at least six vertices.
• Figure 3: Illustration for McConnell's transformation. (a) A circular-arc graph $G$, where edges among the vertices in the clique $K$ (in the shadowed area) are omitted for clarity; (b) a normalized circular-arc model of $G$; (c) the interval graph $G^K$, where edges incident to the universal vertex $s$ are omitted for clarity; and (d) the interval model of $G^K$ derived from (b) by flipping the arcs containing the center of the bottom.
• Figure 4: Interval configurations whose absence in the interval graph $G^K$ asserts condition \ref{['eq:sharp']} in the case when $G$ is $C_4$-free. The square nodes are "in $K$," rhombus "not in $K$," and round "uncertain." Between a square node and a rhombus node, a thick edge is "in $G$," a thin edge is "not in $G$," and it is "uncertain" otherwise (a solid line and a dotted line). There are at least five vertices in \ref{['fig:dag+2e-unlabeled']}, and at least six vertices in \ref{['fig:dag+e-unlabeled']}, \ref{['fig:ddag+e-unlabeled']}, and \ref{['fig:ddag+2e-unlabeled']}.
• Figure 5: (a) An interval graph and (b) its clique path represented as an interval model.

Theorems & Definitions (32)

• Theorem 1.1: lekkerkerker-62-interval-graphs
• Theorem 1.2
• Definition : Annotations
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 2.1
• proof
• Lemma 2.2
• proof