Some relations between the independence number and other parameters in \(K_{1,r}\)-free graphs (Q2764417)

A graph \(G\) is called \(K_{1,r}\)-free, if none of its induced subgraphs is a star \(K_{1,r}\) with \(r\) edges. For such graphs various numerical invariants are studied, namely the independence number \(\alpha(G)\), connectivity \(\kappa(G)\) and toughness \(t(G)\). A graph \(G\) is called \(t\)-tough, if \(|S|\geq t\omega(G-S)\) for every subset \(S\) of the vertex set \(V(G)\) of \(G\) with \(\omega(G- S)\geq 1\); here \(\omega(G- S)\) denotes the number of connected components of the graph obtained from \(G\) by deleting \(S\). The minimum number \(t\) such that \(G\) is \(t\)-tough is the toughness \(t(G)\) of \(G\). The paper contains some inequalities relating these invariants with the numbers of vertices and of edges and with the degrees of the vertices of \(G\).