[Paper Review] Forbidden Induced Subgraphs of Normal Helly Circular-Arc Graphs: Characterization and Detection

Yixin Cao, Luciano N. Grippo+1 more|arXiv (Cornell University)|May 2, 2014

Advanced Graph Theory Research3 references3 citations

TL;DR

This paper provides a complete forbidden induced subgraph characterization for normal Helly circular-arc graphs and presents a linear-time certifying recognition algorithm that outputs a minimal forbidden induced subgraph as a certificate when the input is not a normal Helly circular-arc graph. The approach resolves two open problems posed by Lin, Soulignac, and Szwarcfiter by identifying chordal minimal forbidden subgraphs and enabling efficient detection through structural decomposition and model analysis.

ABSTRACT

A normal Helly circular-arc graph is the intersection graph of arcs on a circle of which no three or less arcs cover the whole circle. Lin, Soulignac, and Szwarcfiter [Discrete Appl. Math. 2013] characterized circular-arc graphs that are not normal Helly circular-arc graphs, and used it to develop the first recognition algorithm for this graph class. As open problems, they ask for the forbidden induced subgraph characterization and a direct recognition algorithm for normal Helly circular-arc graphs, both of which are resolved by the current paper. Moreover, when the input is not a normal Helly circular-arc graph, our recognition algorithm finds in linear time a minimal forbidden induced subgraph as certificate.

Motivation & Objective

• To resolve the open problem of characterizing normal Helly circular-arc graphs by forbidden induced subgraphs.
• To develop a direct, linear-time recognition algorithm for normal Helly circular-arc graphs.
• To provide a minimal forbidden induced subgraph as a negative certificate when the input is not a normal Helly circular-arc graph.
• To extend the understanding of circular-arc graph subclasses by identifying structural obstructions beyond interval and Helly properties.

Proposed method

• The paper identifies chordal minimal forbidden induced subgraphs (FIS) that characterize normal Helly circular-arc graphs, including the long claw, whipping top, and other specific configurations.
• It introduces a certifying recognition algorithm that, in linear time, either constructs a normal Helly circular-arc model or returns a minimal forbidden induced subgraph.
• The method relies on analyzing the structure of the circular-arc model and the neighborhood relationships in the graph, particularly focusing on simplicial vertices and their neighbors.
• A key technique involves traversing the graph’s structure via a chordal graph decomposition and using local checks to detect forbidden subgraphs such as holes or $K_{2,3}$.
• The algorithm uses a procedure to detect minimal forbidden subgraphs by examining vertex pairs and their common neighbors, applying lemmas to verify subgraph minimality.
• It leverages prior results on interval and circular-arc models, particularly the characterization of normal and Helly properties, to restrict the search space for forbidden configurations.

Experimental results

Research questions

• RQ1 What is the complete set of forbidden induced subgraphs that characterize normal Helly circular-arc graphs?
• RQ2 Can a linear-time certifying recognition algorithm be designed for normal Helly circular-arc graphs that provides a minimal forbidden induced subgraph as a certificate for non-membership?
• RQ3 How do the structural properties of normal and Helly circular-arc models interact to define a proper subclass distinct from their intersection?
• RQ4 What are the minimal obstructions that prevent a circular-arc graph from being representable by a normal Helly circular-arc model?
• RQ5 Can the detection of such obstructions be performed efficiently, with linear-time complexity?

Key findings

• The paper identifies a complete set of chordal minimal forbidden induced subgraphs that characterize normal Helly circular-arc graphs, resolving an open problem posed by Lin, Soulignac, and Szwarcfiter.
• A linear-time certifying recognition algorithm is developed that either constructs a normal Helly circular-arc model or returns a minimal forbidden induced subgraph as a certificate.
• The algorithm correctly detects all minimal forbidden subgraphs, including the long claw, whipping top, and $C^*$ configurations, through local neighborhood analysis.
• The method ensures that every non-normal Helly circular-arc graph is rejected with a minimal forbidden induced subgraph, providing a verifiable negative certificate.
• The structural analysis confirms that the class of normal Helly circular-arc graphs is a proper subclass of the intersection of normal and Helly circular-arc graphs, due to the non-existence of models combining both properties in some cases.
• The procedure for detecting forbidden subgraphs runs in $O(n+m)$ time, making the algorithm efficient and suitable for practical implementation.