Total restrained domination in graphs

Authors:
Xing Chen;Juan Liu;Jixiang Meng
Affiliations:
College of Mathematics and Systems Sciences, Xinjiang University, Urumqi, Xinjiang, 830046, PR China and Xinjiang Polytechnical College, Urumqi, Xinjiang, 830091, PR China;College of Mathematics Sciences, Xinjiang Normal University, Urumqi, Xinjiang, 830054, PR China;College of Mathematics and Systems Sciences, Xinjiang University, Urumqi, Xinjiang, 830046, PR China
Venue:
Computers & Mathematics with Applications
Year:
2011

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Abstract

In this paper, we initiate the study of a variation of standard domination, namely total restrained domination. Let G=(V,E) be a graph. A set D@?V is a total restrained dominating set if every vertex in V-D has at least one neighbor in D and at least one neighbor in V-D, and every vertex in D has at least one neighbor in D. The total restrained domination number of G, denoted by @c"t"r(G), is the minimum cardinality of all total restrained dominating sets of G. We determine the best possible upper and lower bounds for @c"t"r(G), characterize those graphs achieving these bounds and find the best possible lower bounds for @c"t"r(G)+@c"t"r(G@?) where both G and G@? are connected.