Results and Problems on Saturation Numbers for Linear Forests

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 sat(n,H) and is called the saturation number of H. In [?], Kászonyi and Tuza determined the values of sat(n,H) when H is a path or a disjoint union of edges. In this paper, we determine the values of sat(n,H) for the disjoint union of paths (a linear forest) within a constant depending only on H . Moreover, we obtain exact values for some special classes and include several conjectures.