On the extendability of quasi-strongly regular graphs with diameter 2 (Q2413638)

A graph \(G\) with an even number of vertices at least \(2l+2\) vertices is \(l\)-extendable if each matching of size \(l\) is contained in a perfect matching of \(G\). A graph \(G\) is quasi-strongly regular with parameters \((n,k,\lambda;\mu_1,\dots,\mu_s)\) if \(G\) is \(k\)-regular on \(n\) vertices, any two adjacent vertices of \(G\) have exactly \(\lambda\) common neighbors, while any two nonadjacent vertices of \(G\) have either \(\mu_1\) or \dots\ or \(\mu_s\) common neighbors. The authors show that a quasi-strongly regular graph with parameters \((n,k,\lambda;\mu_1,\mu_2)\), \(n\geq 6\), \(\mu_1>\mu_2\geq 1\), is not 2-extendable if and only if it is either the complete multipartite graph \(K_{2,2,2}\), or the Möbius ladder on eight vertices, or the Petersen graph, or the lexicographic product \(C_5[2K_1]\).