Let G = (V,E) be a graph. A set S âŠ† V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V âˆ’ S is adjacent to a vertex in V âˆ’ S. A set S âŠ† V is a restrained dominating set if every vertex in V âˆ’ S is adjacent to a vertex in S and to a vertex in V âˆ’ S. The total restrained domination number of G (restrainedâ€¦ (More)

Let G = (V,E) be a graph. A set D âŠ† V is a total outer-connected dominating set of G if D is dominating and G[V âˆ’D] is connected. The total outer-connected domination number of G, denoted Î³tc(G), is the smallest cardinality of a total outer-connected dominating set of G. It is known that if T is a tree of order n â‰¥ 2, then Î³tc(T ) â‰¥ 2n 3 . We will provide aâ€¦ (More)

Let G = (V, E) be a graph. A set S âŠ† V is a restrained dominating set if every vertex in V âˆ’ S is adjacent to a vertex in S and to a vertex in V âˆ’ S. The restrained domination number of G, denoted Î³r (G), is the smallest cardinality of a restrained dominating set of G. We will show that if G is claw-free with minimum degree at least two and G / âˆˆâ€¦ (More)

Let G = (V , E) be a graph. A set S âŠ† V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex in V âˆ’ S is adjacent to a vertex in V âˆ’ S. The total restrained domination number of G, denoted Î³tr (G), is the smallest cardinality of a total restrained dominating set of G. We will show that if G is claw-free,â€¦ (More)