The Minimum Independence Number of a Hasse Diagram

Abstract

Let $P=(X,\le)$ be a finite partially ordered set. That is, $X$ is a finite ground set and $\le$ is a partial ordering on $X$ (a reflexive, transitive, and weakly antisymmetric relation). An $x\in X$ is an immediate predecessor of a $y\in X$ if $x<y$ and there is no $z\in X$ with $x<z<y$ (where $x<y$ means that $x\le y$ and $x\ne y$). The Hasse diagram$H(P)$ is the undirected graph with vertex set $X$ and with $\{x,y\}$ forming an edge if $x$ is an immediate predecessor of $y$ or if $y$ is an immediate predecessor of $x$. We denote bya $\alpha(H(P))$ the independence number of the Hasse diagram, that is, the maximum possible size of a subset $I\subseteq X$ such that no element of $I$ is an immediate predecessor (in $P$) of another element of $I$. This quantity should not be confused with the maximum size of an antichain in $P$, which is sometimes denoted by $\alpha(P)$.