\(\{K_{1,3}, Z_2\}\)-Free Graphs of Low Connectivity

A graph \(G\) is \(\{R, S\}\)-free if \(G\) contains no induced subgraphs isomorphic to \(R\) or \(S\). The graph \(Z_1\) is a triangle with a path of length \(1\) off one vertex; the graph \(Z_2\) is a triangle with a path of length \(2\) off one vertex. A graph that is \(\{K_{1,3}, Z_1\}\)-free is known to be either a cycle or a complete graph minus a matching. In this paper, we investigate the structure of \(\{K_{1,3}, Z_2\}\)-free graphs. In particular, we characterize \(\{K_{1,3}, Z_2\}\)-free graphs of connectivity \(1\) and connectivity \(2\).