# Largest reduced neighborhood clique cover number revisited

## TL;DR

This paper revisits the properties of the largest reduced neighborhood clique cover number in graphs, exploring its bounds, connections to graph density, and conjectures about its behavior in incomparability graphs.

## Contribution

It provides an overview of ${ ilde\beta}_t(G)$ properties, introduces bounded neighborhood clique cover graphs, and proposes conjectures on its exact value and separator theorems.

## Key findings

- Connected ${\hat\beta}_t(G)$ to graph density measures.
- Derived bounds for the neighborhood clique cover number.
- Proposed conjectures on its value and separator properties.

## Abstract

Let $G$ be a graph and $t\ge 0$. The largest reduced neighborhood clique cover number of $G$, denoted by ${\hat\beta}_t(G)$, is the largest, overall $t$-shallow minors $H$ of $G$, of the smallest number of cliques that can cover any closed neighborhood of a vertex in $H$. It is known that ${\hat\beta}_t(G)\le s_t$, where $G$ is an incomparability graph and $s_t$ is the number of leaves in a largest $t-$shallow minor which is isomorphic to an induced star on $s_t$ leaves. In this paper we give an overview of the properties of ${\hat\beta}_t(G)$ including the connections to the greatest reduced average density of $G$, or $\bigtriangledown_t(G)$, introduce the class of graphs with bounded neighborhood clique cover number, and derive a simple lower and an upper bound for this important graph parameter. We announce two conjectures, one for the value of ${\hat\beta}_t(G)$, and another for a separator theorem (with respect to a certain measure) for an interesting class of graphs, namely the class of incomparability graphs which we suspect to have a polynomial bounded neighborhood clique cover number, when the size of a largest induced star is bounded.