The paired bondage number (total restrained bondage number, independent bondage number, k-rainbow bondage number) of a graph G, is the minimum number of edges whose removal from G results in a graph with larger paired domination number (respectively, total restrained domination number, independent domination number, k-rainbow domination number). In this paper we show that the decision problems ...

We show that for every integer k ≥ 2 and every k graphs G1, G2, . . . , Gk, there exists a hull graph with k hull vertices v1, v2, . . . , vk such that link L(vi) = Gi for 1 ≤ i ≤ k. Moreover, every pair a, b of integers with 2 ≤ a ≤ b is realizable as the hull number and geodetic number (or upper geodetic number) of a hull graph. We also show that every pair a, b of integers with a ≥ 2 and b ≥...

MacGillivray and Seyffarth (J. Graph Theory 22 (1996), 213–229) proved that planar graphs of diameter three have domination number at most ten. Recently it was shown (J. Graph Theory 40 (2002), 1–25) that a planar graph of diameter three and of radius two has domination number at most six while every sufficiently large planar graph of diameter three has domination number at most seven. In this ...

‎a total dominating set of a graph $g$ is a set $d$ of vertices of $g$ such that every vertex of $g$ has a neighbor in $d$‎. ‎the total domination number of a graph $g$‎, ‎denoted by $gamma_t(g)$‎, ‎is~the minimum cardinality of a total dominating set of $g$‎. ‎chellali and haynes [total and paired-domination numbers of a tree, akce international ournal of graphs and combinatorics 1 (2004)‎, ‎6...

MacGillivray and Seyffarth (J. Graph Theory 22 (1996), 213–229) proved that planar graphs of diameter two have domination number at most three and planar graphs of diameter three have domination number at most ten. They also give examples of planar graphs of diameter four having arbitrarily large domination numbers. In this paper we improve on their results. We prove that there is in fact a uni...

Let K n denote the Cartesian product Kn Kn Kn, where Kn is the complete graph on n vertices. We show that the domination number of K n is ⌈

The domination number of a graph G = (V,E) is the minimum cardinality of any subset S ⊂ V such that every vertex in V is in S or adjacent to an element of S. Finding the domination numbers of m by n grids was an open problem for nearly 30 years and was finally solved in 2011 by Goncalves, Pinlou, Rao, and Thomassé. Many variants of domination number on graphs, such as double domination number a...

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. We call these graphs total domination critical or just γt-critical. If such a graph G has total domination number k, we call it k-γt-critical. We study an open problem of ...

We introduce the domination search game which can be seen as a natural modiica-tion of the well-known node search game. Various results concerning the domination search number of a graph are presented. In particular, we establish a very interesting connection between domination graph searching and a relatively new graph parameter called dominating target number.