$$(K_{1}\vee {P_{t})}$$ -saturated graphs with minimum number of edges

Abstract

For a fixed graph F, a graph G is F-saturated if G does not contain F as a subgraph, but adding any edge in \(E(\overline{G})\) will result in a copy of F. The minimum size of an F-saturated graph of order n is called the saturation number of F, denoted by sat(n, F). In this paper, we are interested in saturation problem of graph \(K_1\vee {P_t}\) for \(t\ge 2\). As some known results, \(sat(n,K_1\vee {P_t})\) is determined for \(2\le t\le 4\). We will show that \(sat(n,K_1\vee {P_t})=(n-1)+sat(n-1,P_t)\) for \(t\ge 5\) and n sufficiently large. Moreover, \((K_1\vee {P_t})\)-saturated graphs with \(sat(n,K_1\vee {P_t})\) edges are characterized.