Characterization of Chordal Circular-arc Graphs: I. Split Graphs

TL;DR

The paper addresses identifying all minimal split graphs that fail to be circular-arc graphs by linking them to minimal non-interval graphs through McConnell’s $G^K$ transformation, enabling a systematic forbidden-subgraph characterization. It introduces new gadgets and a detailed framework around $H=G^{N[s]}$ and annotated configuration copies to capture exactly when a split graph is not circular-arc, and it provides a linear-time certifying recognition algorithm for split inputs. The work yields a complete set of obstructions and a practical method to certify circular-arc recognition, with significant implications for understanding the structure of circular-arc graphs and their split subclasses. The results offer both theoretical insight and algorithmic tools for efficient recognition and certification in this graph class family.

Abstract

The most elusive problem around the class of circular-arc graphs is identifying all minimal graphs that are not in this class. The main obstacle is the lack of a systematic way of enumerating these minimal graphs. McConnell [FOCS 2001] presented a transformation from circular-arc graphs to interval graphs with certain patterns of representations. We fully characterize these interval patterns for circular-arc graphs that are split graphs, thereby building a connection between minimal split graphs that are not circular-arc graphs and minimal non-interval graphs. This connection enables us to identify all minimal split graphs that are not circular-arc graphs. As a byproduct, we develop a linear-time certifying recognition algorithm for circular-arc graphs when the input is a split graph.

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 chordal graphs that are not interval graphs.
• Figure 3: The complements of $k$-suns, for $k = 3, 4, 5$.
• Figure 4: The Venn diagram of the four graph classes. The blue and red areas consists of minimal forbidden induced subgraphs of the class of Helly circular-arc graphs and the class of circular-arc graphs, respectively.
• Figure 5: Minimal split graphs that are not interval graphs.

Theorems & Definitions (53)

• Theorem 1.1: lekkerkerker-62-interval-graphs
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3