On Computational and Combinatorial Properties of the Total Co-independent Domination Number of Graphs. (13th April 2018) - Cabrera Martínez, Abel; Hernández Mira, Frank A; Sigarreta Almira, José M; Yero, Ismael G
Abstract: 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 γ t, c o i ( G ) . In this work we study some complexity and combinatorial properties of γ t, c o i ( G ) . Specifically, we prove that deciding whether γ t, c o i ( G ) ≤ k for a given integer k is an NP-complete problem and give several bounds on γ t, c o i ( 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.