Saturation Numbers for Linear Forests $P_7+tP_2$

TL;DR

This paper determines the minimum number of edges in graphs that are saturated with respect to a linear forest consisting of a path of length 7 and t disjoint edges, for sufficiently large n.

Contribution

The paper explicitly calculates the saturation number for the linear forest $P_7 + tP_2$ and characterizes extremal graphs for large n.

Findings

Exact saturation number for $P_7 + tP_2$ when $n o ext{large}$

Characterization of extremal graphs achieving this saturation number

Bounds on n for the validity of the results

Abstract

Let $H$ be a fixed graph, a graph G is $H$-saturated if it has no copy of $H$ in $G$, but the addition of any edge in $E(\overline G)$ to $G$ results in an $H$-subgraph. The saturation number sat$(n,H)$ is the minimum number of edges in an $H$-saturated graph on $n$ vertices. In this paper, we determine the saturation number sat$(n,P_7+tP_2)$ for $n\geq \frac {14}{5}t+27$ and characterize the extremal graphs for $n\geq \frac{14}{13}(3t+25)$.