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+6)/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 ∈{2,3,4}.