Saturation Numbers for Linear Forests $P_6$ + $tP_2$

TL;DR

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

Contribution

It explicitly calculates the saturation number for the graph P_6 + tP_2 and characterizes extremal graphs for large n.

Findings

Exact saturation number for P_6 + tP_2 when n ≥ 10t/3 + 10

Characterization of extremal graphs for n > 10t/3 + 20

Abstract

A graph $G$ is $H$-saturated if it contains no $H$ as a subgraph, but does contain $H$ after the addition of any edge in the complement of $G$. The saturation number, $sat (n, H)$, is the minimum number of edges of a graph in the set of all $H$-saturated graphs with order $n$. In this paper, we determine the saturation number $sat (n, P_6 + tP_2)$ for $n \geq 10t/3 + 10$ and characterize the extremal graphs for $n >10t/3 + 20$.