Total restrained domination in claw-free graphs with minimum degree at least two

Let G=(V,E)G=(V,E) be a graph. A set S⊆VS⊆V is a total restrained dominating set if every vertex is adjacent to a vertex in SS and every vertex in V−SV−S is adjacent to a vertex in V−SV−S. The total restrained domination number of GG, denoted γtr(G)γtr(G), is the smallest cardinality of a total restrained dominating set of GG. We will show that if GG is claw-free, connected, has minimum degree at least two and GG is not one of nine exceptional graphs, then γtr(G)≤4n7.

► Let G=(V,E)G=(V,E) be a graph.
► A set S⊆VS⊆V is a total restrained dominating set if every vertex is adjacent to a vertex in SS and every vertex in V−SV−S is adjacent to a vertex in V−SV−S.
► The total restrained domination number of GG, denoted γtr(G)γtr(G), is the smallest cardinality of a total restrained dominating set of GG.
► We will show that if GG is claw-free, connected, has minimum degree at least two and GG is not one of nine exceptional graphs, then γtr(G)≤4n7.