Concetti Chiave
Circular-arc graphs, a class of intersection graphs, may not satisfy the Helly property in all their intersection models. The authors investigate the Helly properties of circular-arc graphs, providing algorithms to determine the Helly status of individual cliques and to solve the Helly Cliques problem.
Sintesi
The paper focuses on the Helly properties of circular-arc graphs, which are defined as intersection graphs of arcs of a fixed circle. Unlike interval graphs and chordal graphs, whose intersection models always satisfy the Helly property, circular-arc graphs may have cliques that are Helly in some but not all of their intersection models.
The authors first provide an alternative proof of a theorem by Lin and Szwarcfiter, which states that for every circular-arc graph, either every normalized model satisfies the Helly property or no normalized model does. They then study the Helly properties of individual cliques in circular-arc graphs, classifying them as always-Helly, always-non-Helly, or ambiguous, and provide a polynomial-time algorithm to determine the type of a given clique.
The authors also investigate the Helly Cliques problem, where given a circular-arc graph and some of its cliques, the task is to determine if there exists an intersection model in which all the specified cliques satisfy the Helly property. They show that Helly Cliques is FPT when parameterized by the number of cliques, provide a lower bound under the Exponential Time Hypothesis, and give a polynomial kernel for the problem.
The results in this paper have applications in the recognition of H-graphs, a generalization of various geometric intersection graph classes.