On some subclasses of interval catch digraphs

Abstract

A digraph G = ( V , E ) is an interval catch digraph if for each vertex v ∈ V , one can associate an interval on real line and a point within it (say ( I v , p v ) ) in such a way that u v ∈ E if and only if p v ∈ I u . It was introduced by Maehara in 1984. It has many applications in real world situations like networking and telecommunication. In his introducing paper Maehara proposed a conjecture for the characterization of central interval catch digraph (where p v is the mid-point I v for each v ∈ V ) in terms of forbidden subdigraphs. In this paper, we disprove the conjecture by showing counter examples. Also we characterize this digraph by defining a suitable mapping from the vertex set to the real line. We study oriented interval catch digraphs and characterize an interval catch digraph when it is a tournament. Finally, we characterize a proper interval catch digraph and establish relationships between these digraph classes.