Results and problems on saturation numbers for linear forests (Q2799692)

A graph \(G\) is called \(H\)-saturated if \(G\) contains no copy of \(H\), but for any edge \(e\) in the complement of \(G\), the graph \(G+e\) contains some copy of \(H\). The minimum size of an \(n\)-vertex \(H\)-saturated graph is denoted by \(\mathrm{sat}(n,H)\) and is called the saturation number of \(H\). The maximum size of an \(n\)-vertex \(H\)-saturated graph is the well known Turán extremal number \(\mathrm{ex}(n,H)\). In the paper, the values of \(\mathrm{sat}(n, H)\) for the disjoint union of paths within a constant are determined. Moreover, some exact values for some special classes are obtained and several related conjectures are posed.