Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Distance domination and distance irredundance in graphs. | Distance domination and distance irredundance in graphs Adriana Hansberg Dirk Meierling and Lutz Volkmann Lehrstuhl II fur Mathematik RWTH Aachen University 52056 Aachen Germany e-mail hansberg meierling volkm @ Submitted Feb 13 2007 Accepted Apr 25 2007 Published May 9 2007 Mathematics Subject Classihcation 05C69 Abstract A set D c V of vertices is said to be a connected distance k-dominating set of G if the distance between each vertex u 2 V D and D is at most k and D induces a connected graph in G . The minimum cardinality of a connected distance k-dominating set in G is the connected distance k-domination number of G denoted by yk G yk G respectively . The set D is dehned to be a total k -dominating set of G if every vertex in V is within distance k from some vertex of D other than itself. The minimum cardinality among all total k-dominating sets of G is called the total k-domination number of G and is denoted by yk G . For x 2 X c V if Nk x - Nk X - x the vertex x is said to be k-irredundant in X. A set X containing only k-irredundant vertices is called k -irredundant. The k-irredundance number of G denoted by irk G is the minimum cardinality taken over all maximal k-irredundant sets of vertices of G. In this paper we establish lower bounds for the distance k-irredundance number of graphs and trees. More precisely we prove that Ạ 1 irk G yk G 2k for each connected graph G and 2k 1 irk T yk T 2k VI 2k - kni T for each tree T V E with n1 T leaves. A class of examples shows that the latter bound is sharp. The second inequality generalizes a result of Meierling and Volkmann 9 and Cyman Lemanska and Raczek 2 regarding yk and the hrst generalizes a result of Favaron and Kratsch 4 regarding ir 1. Furthermore we shall show that yk G 3k 1 yk G 2k for each connected graph G thereby generalizing a result of Favaron and Kratsch 4 regarding k 1. Keywords domination irredundance distance domination number total domination number connected domination .
