On total covers of graphs (Q1198645)

A total cover of a graph \(G\) is a subset of \(V(G)\cup E(G)\) which covers all elements of \(V(G)\cup E(G)\). The total covering number \(\alpha_ 2(G)\) of a graph \(G\) is the minimum cardinality of a total cover in \(G\). In \textit{Y. Alavi}, \textit{M. Behzad}, \textit{L. M. Lesniak-Foster} and \textit{E. A. Nordhaus} [J. Graph Theory 1, 135-140 (1977; Zbl 0376.05045)] it was shown that \(\alpha_ 2(G)\leq (n+1)/2\) for a connected graph \(G\) of order \(n\). Here the authors consider those graphs \(G\) with \(\alpha_ 2(G)=(n+1)/2\). Among other things they show that such a graph with even order has a 1-factor and with odd order is factor-critical.