Saturation numbers of $K_{2}\vee P_{k}$

TL;DR

This paper determines the minimum number of edges in graphs that are saturated with respect to the graph $K_2 abla P_k$, providing exact formulas and characterizations for various cases, and suggesting directions for future research.

Contribution

It offers a precise formula for the saturation number of $K_2 abla P_k$ and characterizes the extremal saturated graphs, advancing understanding of saturation in graph theory.

Findings

Established that $sat(n, K_2 abla P_k) = 2n - 3 + sat(n-2, P_k)$ for large $n$ and $k ext{,}$

Characterized the structure of saturated graphs for specific small values of $k$,

Proposed open questions for further exploration in saturation problems.

Abstract

A graph $G$ is called $H$-saturated if $G$ contains no copy of $H$, but $G+e$ contains a copy of $H$ for any edge $e\in E(\overline{G})$. The saturation number of $H$ is the minimum number of edges in an $H$-saturated graph of order $n$, denoted by $sat(n,H)$. In this paper, we investigate $sat(n,K_{2}\vee P_{k})$, where $k\geq 3$. Let $a_k$ be an integer, defined as follows: $a_k=k$ for $3\leq k\leq 5$; $a_k=3\cdot 2^{t-1}-2$ for $k=2t\geq 6$; and $a_k=2^{t+1}-2$ for $k=2t+1\geq 7$. We show that $sat(n, K_{2}\vee P_{k})=2n-3+sat(n-2,P_{k})$ for $n\geq a_k+2$ and $k\geq 3$, characterize the $K_{2}\vee P_{k}$-saturated graphs with $sat(n,K_{2}\vee P_{k})$ edges, the $K_{1}\vee P_{k}$-saturated graphs with $sat(n,K_{1}\vee P_{k})$ edges for $3\leq k\leq5$ and the $P_{k}$-saturated graphs with $sat(n, P_{k})$ edges for $3\leq k\leq4$. Furthermore, we propose some questions for further research.