Domination alteration sets in graphs

Abstract

The domination number a(G) of a graph G is the size of a minimum dominating set, i.e., a set of points with the property that every other point is adjacent to a point of the set. In general a(G) can be made to increase or decrease by the removal of points from G. Our main objective is the study of this phenomenon.For example we show that if T is a tree with at least three points then a(T -u) > a(T) if and only if u is in every minimum dominating set of 7'. Removal of a set of lines from a graph G cannot decrease the domination number. We obtain some upper bounds on the size of a minimum set of lines which when removed from G increases the domination number.

Some Progress on Total Bondage in Graphs

“…This concept was introduced by Bauer, Harary, Nieminen and Suffel in [1], and has been further studied for example in [4,5,13]. For more information on this topic we refer the reader to the survey article by Dunbar, Haynes, Teschner and Volkmann [3].…”

“…Let A be the set of all pendant edges incident with x 3 . Let T 1 be a graph obtained from T − x 1 x 2 by removing each edge incident with x 3 with exception of x 2 x 3 and the edges in A. Let H 1 be the component of T 1 containing x 3 .…”

“…k and T 2 k are two trees belonging to R k and let x 1 1 , x 2 1 be vertices with label B belonging to…”

О Числе Зависимости Медиального Графа

“…Понятие числа зависимости ввели Бауэр, Харари, Ниеминен и Сюффель в [2]; это число исследовали многие авторы -Финк, Джекобсон, Кинч и Робертс [3], Хартнелл и Ролл [4] и другие. Число зависимости b(G) непустого графа G определяется как мощность наименьшего множества ребер, удаление которого из G приводит к увеличению числа доминирования.…”

Bondage number of mesh networks

“…Therefore, it is of significance to develop a technique to determine bondage numbers for some special graphs or networks. However, the exact value of the bondage number has been determined for only a few classes of graphs, such as complete graphs, paths, cycles and complete t-partition graphs (see, Fink et al [8] for the undirected cases, Huang and Xu [14], Zhang et al [31] for the directed cases), trees (see, Bauer et al [1], Hartnell and Rall [10], Hartnell et al [9], Topp and Vestergaard [28], Teschner [26]), de Bruijn and Kautz digraphs (see, Huang and Xu [14]). …”