Threshold numbers of some graphs (2406.12063v3)

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.