Abstract
A set S of vertices in graph G=(V,E) is a total, restrained, dominating, set, abbreviated TRDS, of G if every vertex of G is adjacent to a vertex in S and every vertex of V-S is adjacent to a vertex in V-S. The total, restrained, domination, number of G, denoted by gamma _{tr}(G), is the minimum cardinality of a TRDS of G. Jiang and Kang (J Comb Optim. 19:60–68, 2010) characterized the connected claw-free graph G of order n with gamma _{tr}(G)=n. This paper studies the total restrained domination number of claw-free graphs and characterizes the connected claw-free graph G of order n with gamma _{tr}(G)=n-2.
Highlights
Ŵ122 is a collection of all graphs obtained from G′ ∈ Ŵ1 by adding a new vertex and joining it to each endpoint of a pendant edge of G′
Let G = (V, E) be a simple graph with vertex set V and edge set E
The study focuses on the total restrained domination in claw-free graphs
Summary
Ŵ122 is a collection of all graphs obtained from G′ ∈ Ŵ1 by adding a new vertex and joining it to each endpoint of a pendant edge of G′. Theorem 1 Let G be a connected claw-free graph of order n ≥ 4. Let G be a connected claw-free graph of order n with γtr(G) = n − 2 and S be a γtr-set of G.
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