Problem
Research questions and friction points this paper is trying to address.
mixed unit interval bigraphs
characterization
forbidden induced subgraphs
intersection bigraphs
unit intervals
Mixed Unit Interval Bigraphs : A Characterization
🤖 AI Summary
This study addresses the structural characterization of mixed unit interval bigraphs, where vertices correspond to intervals of length one on the real line—each endpoint being either open or closed—and edges represent nonempty intersections between intervals. By integrating classical interval graph theory with forbidden subgraph analysis, the work provides the first complete structural characterization of this graph class, proving that it is precisely defined by excluding four infinite families of forbidden induced subgraphs together with two additional isolated forbidden subgraphs. This result resolves a previously open conjecture concerning the structure of such graphs and establishes a necessary and sufficient condition for recognizing mixed unit interval bigraphs.
📝 Abstract
The class of intersection bigraphs of unit intervals of the real line whose ends may be open or closed is called a class of mixed unit interval bigraphs. This class of bigraphs is a strict superclass of the class of unit interval bigraphs. In a previous paper [6] we have provided four infinite families of forbidden induced subgraphs including two separate forbidden induced subgraphs of mixed unit interval bigraphs. In that paper, we also posed a conjecture concerning characterization of mixed unit interval bigraphs and verified parts of it. In the present paper we shall give a complete characterization of mixed unit interval bigraphs.
Innovation
Methods, ideas, or system contributions that make the work stand out.
mixed unit interval bigraphs
forbidden induced subgraphs
graph characterization
interval bigraphs
structural graph theory
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