A Note on Threshold Dimension of Permutation Graphs

TL;DR

This paper establishes bounds on the threshold dimension of permutation graphs, showing it is at most the size of the maximum independent set and at most half the number of vertices, with both bounds being tight.

Contribution

The paper proves that the threshold dimension of permutation graphs is bounded above by the maximum independent set size and by half the number of vertices, providing tight bounds.

Findings

Threshold dimension of permutation graphs is at most the independence number.

Threshold dimension is at most half the number of vertices in permutation graphs.

Both bounds are proven to be tight.

Abstract

A graph $G(V,E)$ is a threshold graph if there exist non-negative reals $w_v, v \in V$ and $t$ such that for every $U \subseteq V$, $\sum_{v \in U} w_v\leq t$ if and only if $U$ is a stable set. The {\it threshold dimension} of a graph $G(V,E)$, denoted as $t(G)$, is the smallest integer $k$ such that $E$ can be covered by $k$ threshold spanning subgraphs of $G$. A permutation graph is a graph that can be represented as the intersection graph of a family of line segments that connect two parallel lines in the Euclidean plane. In this paper we will show that if $G$ is a permutation graph then $t(G) \leq \alpha(G)$ (where $\alpha(G)$ is the cardinality of maximum independent set in $G$) and this bound is tight. As a corollary we will show that $t(G) \leq \frac{n}{2}$ where $n$ is the number of vertices in the permutation graph $G$. This bound is also tight.