The Edge Spectrum of \(K_{4}\)-Saturated Graphs

Any \( H \)-free graph \( G \) is called \( H \)-saturated if the addition of any edge \( e \notin E(G) \) results in \( H \) as a subgraph of \( G \). The minimum size of an \( H \)-saturated graph on \( n \) vertices is denoted by \( sat(n, H) \). The edge spectrum for the family of graphs with property \( P \) is the set of all sizes of graphs with property \( P \). In this paper, we find the edge spectrum of \( K_4 \)-saturated graphs. We also show that if \( G \) is a \( K_4 \)-saturated graph, then either \( G \cong K_{1,1,n-2} \) or \( \delta(G) \geq 3 \), and we detail the exact structure of a \( K_4 \)-saturated graph with \( \kappa(G) = 2 \) and \( \kappa(G) = 3 \).