Abstract
A set S of vertices in a graph G=(V,E) is a total restrained dominating set (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 γ tr (G), is the minimum cardinality of a TRDS of G. In this paper we characterize the claw-free graphs G of order n with γ tr (G)=n. Also, we show that γ tr (G)≤n−Δ+1 if G is a connected claw-free graph of order n≥4 with maximum degree Δ≤n−2 and minimum degree at least 2 and characterize those graphs which achieve this bound.
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Research was partially supported by the National Nature Science Foundation of China (Nos. 10571117, 60773078) and the ShuGuang Plan of Shanghai Education Development Foundation (No. 06SG42).
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Jiang, H., Kang, L. Total restrained domination in claw-free graphs. J Comb Optim 19, 60–68 (2010). https://doi.org/10.1007/s10878-008-9161-1
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DOI: https://doi.org/10.1007/s10878-008-9161-1