Minimizing the number of edges in (Pk ∪ K3)-saturated connected graphs

Abstract

For a graph H, a graph G is H-saturated if it contains no copy of H as a (not necessarily induced) subgraph, but the addition of any edge missing from G creates a copy of H in the resultant graph. The connected saturation number sat′(n, H) is defined as the minimum number of edges in H-saturated connected graphs on n vertices. In this paper we consider the (Pk ∪ K3)-saturated connected graphs on n vertices and focus on the determination of sat′(n, Pk ∪ K3). We prove that n + 2 ≤ sat′(n, Pk ∪ K3) ≤ n + 3k-6/2 for n > 3k+4/2 with k ≥ 4 and characterize the extremal graphs at which the upper bounds are attained. Moreover, the exact values of sat′(n, Pk ∪ K3) are determined with k ∈ {3, 4} and we get sat(n, P2 ∪ K3) = sat′(n, P2 ∪ K3) = n.