Upper bounds for domination related parameters in graphs on surfaces

AbstractIn this paper we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic. We also present upper bounds for the restrained bondage number, the total restrained bondage number and the restricted edge connectivity of graphs in terms of the orientable/nonorientable genus and maximum degree

New bounds on the signed total domination number of graphs

In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turan, we present a sharp lower bound on this parameter for graphs with no complete graph of order r+1 as a subgraph. Also, we prove that n-2(s-s') is an upper bound on the signed total domination number of any tree of order n with s support vertices and s' support vertives of degree two. Moreover, we characterize all trees attainig this bound.Comment: This paper contains 11 pages and one figur

On upper bounds for total kk-domination number via the probabilistic method

summary:For a fixed positive integer $k$ and $G=(V, E)$ a connected graph of order $n$, whose minimum vertex degree is at least $k$, a set $S\subseteq V$ is a total $k$-dominating set, also known as a $k$-tuple total dominating set, if every vertex $v\in V$ has at least $k$ neighbors in $S$. The minimum size of a total $k$-dominating set for $G$ is called the total $k$-domination number of $G$, denoted by $\gamma_{kt}(G)$. The total $k$-domination problem is to determine a minimum total $k$-dominating set of $G$. Since the exact problem is in general quite difficult to solve, it is also of interest to have good upper bounds on the total $k$-domination number. In this paper, we present a probabilistic approach to computing an upper bound for the total $k$-domination number that improves on some previous results