On some subclasses of circular-arc catch digraphs

TL;DR

This paper introduces circular-arc catch digraphs, characterizes proper subclasses, and explores their properties and relations, expanding the theory of intersection digraphs with applications in network and telecommunication fields.

Contribution

It defines and characterizes a new class of catch digraphs called circular-arc catch digraphs and their proper subclasses, providing structural insights and forbidden subdigraph characterizations.

Findings

Proper circular-arc catch digraphs are characterized by monotone circular ordering.

Underlying graphs of proper oriented circular-arc catch digraphs are proper circular-arc graphs.

Certain subclasses of these digraphs are characterized by forbidden subdigraphs.

Abstract

Catch digraphs was introduced by Hiroshi Maehara in 1984 as an analog of intersection graphs where a family of pointed sets represents a digraph. After that Prisner continued his research particularly on interval catch digraphs by characterizing them diasteroidal triple free. It has numerous applications in the field of real world problems like network technology and telecommunication operations. Recently, we characterized three important subclasses of interval catch digraphs. In this article we introduce a new class of catch digraphs, namely circular-arc catch digraphs. The definition is same as interval catch digraph, only the intervals are replaced by circular-arcs here. We present the characterization of proper circular-arc catch digraphs, which is a natural subclass of circular-arc catch digraphs where no circular-arc is contained in other properly. For this we introduce a concept, namely monotone circular ordering for the vertices of the augmented adjacency matrix of it. Next we find that underlying graph of a proper oriented circular-arc catch digraph is a proper circular-arc graph. Also we characterize proper oriented circular-arc catch digraphs by defining a certain kind of circular vertex ordering of its vertices. Another interesting result is to characterize oriented circular-arc catch digraphs which are tournaments in terms of forbidden subdigraphs. Further we study some properties of an oriented circular-arc catch digraph. In conclusion we discuss the relations between these subclasses of circular-arc catch digraphs.