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. Let G be a cubic graph of order n. In this paper we establish an upper bound on γ tr (G). If adding the restriction that G is claw-free, then we show that γ tr (G) = γ t (G) where γ t (G) is the total domination number of G, and thus some results on total domination in claw-free cubic graphs are valid for total restrained domination.
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Research was partially supported by the NNSF of China (Nos. 60773078, 10832006), the ShuGuang Plan of Shanghai Education Development Foundation (No. 06SG42) and Shanghai Leading Academic Discipline Project (No. S30104).
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Jiang, H., Kang, L. & Shan, E. Total Restrained Domination in Cubic Graphs. Graphs and Combinatorics 25, 341–350 (2009). https://doi.org/10.1007/s00373-009-0854-8
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DOI: https://doi.org/10.1007/s00373-009-0854-8