Total restrained domination in graphs with minimum degree two

Abstract

In this paper, we continue the study of total restrained domination in graphs, a concept introduced by Telle and Proskurowksi (Algorithms for vertex partitioning problems on partial k-trees, SIAM J. Discrete Math. 10 (1997) 529-550) as a vertex partitioning problem. A set S of vertices in a graph G = (V , E) is a total restrained dominating set of G if every vertex is adjacent to a vertex in S and every vertex of V \S is adjacent to a vertex in V \S. The minimum cardinality of a total restrained dominating set of G is the total restrained domination number of G, denoted by tr (G). Let G be a connected graph of order n with minimum degree at least 2 and with maximum degree where n − 2. We prove that if n 4, then tr (G) n − 2 − 1 and this bound is sharp. If we restrict G to a bipartite graph with 3, then we improve this bound by showing that tr (G) n − 2 3 − 2 9 √ 3 − 8 − 7 9 and that this bound is sharp.

On Roman, Global and Restrained Domination in Graphs

“…Theorem 1 implies that when δ(G) is large, γ(G)/n is close to 0 for any graph G. Similar results were proved for the global and Roman domination numbers in the previous sections. However, for the total restrained domination numbers this is not the case, because for any δ there exists (see [8]) an infinite family of graphs G with minimum degree δ, for which γ tr (G)/n → 1 when n tends to ∞. The above is also true for the restrained domination number.…”

An Improved Upper Bound on the Total Restrained Domination Number in Cubic Graphs

“…The total domination number of G, denoted by γ t (G), is the minimum cardinality of a TDS of G, while the total restrained domination number of G, denoted by γ tr (G), is the minimum cardinality of a TRDS of G. Total domination in graphs is very well studied in graph theory (see, for example, the recent papers [8,[10][11][12]19]). Recent papers on total restrained domination in graphs can be found, for example, in [2,5,6,9,13,[15][16][17]21].…”

Restrained and Total Restrained Domination of Ladder Graphs

“…Similarly, we can answer many such situations using the concepts of domination. For more survey work on total restrained domination, we refer Cockayne et al [1], Cyman and Raczek [2], Hattingh et al [9][10][11], Haynes et al [12], Henning and Martiz [14], Raczek and Cyman [15], and Telle and Proskurowski [16].…”