Clique Transversal-Critical, Fixed, Free and Totally Free Elements

A vertex v clique dominates a clique l if v is incident on l. A set D ⊆ V is a clique transversal set if every clique in G is clique dominated by some vertex in D. The clique transversal number τc = τc(G) is the cardinality of a minimum clique transversal set of G. This paper explores properties of vertices and edges based on their membership in all, at least one but not all, or none of the clique transversal sets. A graph G is defined as τc-dot-critical if contracting any edge reduces the clique transversal number. We establish bounds for τc-dotcritical graphs and a lower bound for the full open domination number of a graph in terms of the maximum signature.