Totally equimatchable graphs
Let \(G=(V,E)\) be a connected graph. The elements of the set \(V(G)\cup E(G)\) are called elements of \(G\). A set \(M\) of elements of \(G\) is called a total matching if the elements of \(M\) are pairwise independent, and \(G\) is totally equimatchable if every maximal total matching of \(G\) is maximum. The authors characterize totally equimatchable graphs. When complete graphs are denoted by \(K_n\), complete bipartite graphs by \(K_{n,n}\) and complete windmills by \(K_1+\bigcup^n_{i=1} K_{2m_i}\), where \(K_1+\bigcup^n_{i=1} K_{2m_i}\) is the graph obtained from the disjoint union of \(K_1\), \(K_{2m_1},\dots,K_{2m_n}\) by joining the only vertex of \(K_1\) to every vertex in \(\bigcup^n_{i=1} V(K_{2m_i})\), they prove as main result that a connected graph is totally equimatchable iff it is one of the graphs \(K_n\), \(K_{n,n}\), and \(K_1+\bigcup^n_{i=1} K_{2m_i}\), where \(n\) and \(m_1,\dots,m_n\) are any positive integers. Moreover, the precise values of certain parameters and their interrelations are given.
scientific article; zbMATH DE number 1019073