Independent Chains in Acyclic Posets

TL;DR

This paper precisely determines the maximum size of induced vertex-disjoint unions of cliques in comparability graphs of acyclic posets, revealing a logarithmic relationship with the size of the poset.

Contribution

It provides an exact value for the parameter in the class of comparability graphs of acyclic posets, a problem previously unresolved.

Findings

The parameter a(G) is approximately (n+o(n))/log2(n) for acyclic poset comparability graphs.

The paper establishes a tight bound for the maximum induced union of cliques in this class.

It advances understanding of the structure of comparability graphs of acyclic posets.

Abstract

We consider the problem of determining the maximum order of an induced vertex-disjoint union of cliques in a graph. More specifically, given some family of graphs $\mathcal{G}$ of equal order, we are interested in the parameter $a(\mathcal{G}) = \min_{G \in \mathcal{G}} \max \{ |U| : U \subseteq V, G[U] \text{ is a vertex-disjoint union of cliques} \}$. We determine the value of this parameter precisely when $\mathcal{G}$ is the family of comparability graphs of $n$-element posets with acyclic cover graph. In particular, we show that $a(\mathcal{G}) = (n+o(n))/\log_2 (n)$ in this class.