Saturation Numbers for Linear Forests $P_7+t{P}_2$

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$(n,P_7+t{P}_2)$ for $n\ge \frac{14}{5}t+27$ and characterize the extremal graphs for $n\ge \frac{14}{13}(3t+25)$.

11 pages