Threshold numbers of some graphs

TL;DR

This paper introduces the concept of threshold numbers in graphs, determines these numbers for various path-related graphs, and extends existing results to more complex graph classes, solving an open problem.

Contribution

It defines threshold numbers for graphs, computes these for path-related graphs, and extends prior work to new graph classes, addressing an open problem.

Findings

Threshold numbers for linear forests, ladders, and tents are determined.

Exact threshold numbers for specific multipartite and cluster graphs are established.

Results solve an open problem on threshold numbers for certain graph classes.

Abstract

A graph $G=(V,E)$ is called a \emph{$k$-threshold graph} with \emph{thresholds} $\theta_1<\theta_2<...<\theta_k$ if we can assign a real number $r(v)$ to each vertex $v\in V$, such that for any $u,v\in V$, we have $uv\in E$ if and only if $r(u)+r(v)\ge \theta_i$ holds true for an odd number of elements in $\{\theta_1,\theta_2,...,\theta_k\}$. The smallest integer $k$ such that $G$ is a $k$-threshold graph is called the \emph{threshold number} of $G$. For the complete multipartite graphs and the cluster graphs, Kittipassorn and Sumalroj determined the exact threshold numbers of $K_{n\times 3}$ and $nK_3$. In this paper, first we determine the threshold numbers of some path-related graphs, including linear forests, ladders, and tents. Then, on the basis of Kittipassorn and Sumalroj's results, we determine the exact threshold numbers of $K_{n_1\times 1, n_2\times 2, n_3\times 3}$ and $n_1 K_1\cup n_2 K_2\cup n_3 K_3$, which solve a problem proposed by Sumalroj.