On Computational and Combinatorial Properties of the Total Co-independent Domination Number of Graphs

A subset D of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex of D. The total dominating set D is called a total co-independent dominating set if the subgraph induced by V-D is edgeless and has at least one vertex. The minimum cardinality of any total co-independent dominating set is the total co-independent domination number of G and is denoted by gamma(t,coi) (G). In this work we study some complexity and combinatorial properties of gamma(t,coi) (G). Specifically, we prove that deciding whether gamma(t,coi) (G) <= k for a given integer k is an NP-complete problem and give several bounds on gamma(t,coi) (G). Moreover, since any total co-independent dominating set is a total dominating set, we characterize all the trees having equal total co-independent domination number and total domination number.