Intersection graph of maximal stars

TL;DR

This paper studies the properties and recognition of star graphs, which are intersection graphs of maximal stars in a graph, providing bounds, characterizations, and classifications for these structures.

Contribution

It introduces a quadratic bound on star-critical pre-images, offers a Krausz-type characterization, and classifies small star graphs, advancing understanding of their structure and recognition.

Findings

Star graphs are biconnected.

Every edge in a star graph belongs to at least one triangle.

The diameter of a star graph is bounded by the diameter of its pre-image.

Abstract

A biclique of a graph $G$ is an induced complete bipartite subgraph of $G$ such that neither part is empty. A star is a biclique of $G$ such that one part has exactly one vertex. The star graph of $G$ is the intersection graph of the maximal stars of $G$. A graph $H$ is star-critical if its star graph is different from the star graph of any of its proper induced subgraphs. We begin by presenting a bound on the size of star-critical pre-images by a quadratic function on the number of vertices of the star graph, then proceed to describe a Krausz-type characterization for this graph class; we combine these results to show membership of the recognition problem in \textsf{NP}. We also present some properties of star graphs. In particular, we show that they are biconnected, that every edge belongs to at least one triangle, characterize the structures the pre-image must have in order to generate degree two vertices, and bound the diameter of the star graph with respect to the diameter of its pre-image. Finally, we prove a monotonicity theorem, which we apply to list every star graph on at most eight vertices.