Aneta Dudek , A . Paweł Wojda P m - SATURATED GRAPHS WITH MINIMUM SIZE

By Pm we denote a path of order m. A graph G is said to be Pm − saturated if G has no subgraph isomorphic to Pm and adding any new edge to G creates a Pm in G. In 1986 L. Kászonyi and Zs. Tuza considered the following problem: for given m and n find the minimum size sat(n;Pm) of Pm-saturated graph and characterize the graphs of Sat(n; Pm) – the set of Pm-saturated graphs of minimum size. They have solved this problem for n ≥ am where am = { 3 · 2 − 2 if m = 2k, k > 2 2 − 2 if m = 2k + 1, k ≥ 2 . We define bm = { 3 · 2 if m = 2k, k ≥ 3 3 · 2 − 1 if m = 2k + 1, k ≥ 3 and give sat(n; Pm) and Sat(n; Pm) for m ≥ 6 and bm ≤ n < am.