The saturation number of $K^s_t$

TL;DR

This paper determines the saturation number for specific virus graphs, expanding understanding of minimal edges in graphs avoiding these substructures, and characterizes the extremal saturated graphs.

Contribution

It provides exact saturation numbers and structural descriptions for $K^3_3$ and $K^2_t$ with $t\u2265 4$, extending prior results on $K^2_3$.

Findings

Calculated $sat(n,K^3_3)$ and $sat(n,K^2_t)$ for $t\u2265 4$.

Characterized the extremal graphs achieving these saturation numbers.

Abstract

For a given graph $F$, a graph $G$ is said to be $F$-saturated if $G$ contains no copy of $F$ but for any edge $uv\notin E(G)$, $G+uv$ contains a copy of $F$. The saturation number $sat(n,F)$ is defined as the minimum number of edges among all $n$-vertex $F$-saturated graphs. The virus graph $K^s_t$, where $s\geq0$ and $t\geq \max\{3,s\}$, is a graph of order $s+t$ constructed by attaching $s$ distinct leaves to $s$ different vertices of a complete graph $K_t$. Hua and Peng [Discrete Math. 349 (2026) 114674] determined $sat(n,K^2_3)$ and characterized its corresponding extremal graphs. In this paper, we determine $sat(n,K^3_3)$ and $sat(n,K^2_t)$ with $t\geq 4$, together with the structural descriptions of the related extremal saturated graphs.