Further results on packing related parameters in graphs

Given a graph G = (V, E), a set B subset of V (G) is a packing in G if the closed neighborhoods of every pair of distinct vertices in B are pairwise disjoint. The packing number rho(G) of G is the maximum cardinality of a packing in G. Similarly, open packing sets and open packing number are defined for a graph G by using open neighborhoods instead of closed ones. We give several results concerning the (open) packing number of graphs in this paper. For instance, several bounds on these packing parameters along with some Nordhaus-Gaddum inequalities are given. We characterize all graphs with equal packing and independence numbers and give the characterization of all graphs for which the packing number is equal to the independence number minus one. In addition, due to the close connection between the open packing and total domination numbers, we prove a new upper bound on the total domination number gamma(t)(T) for a tree T of order n >= 2 improving the upper bound gamma(t)(T) <= (n + s)/2 given by Chellali and Haynes in 2004, in which s is the number of support vertices of T

Domination and forcing

This thesis comprises the results of five research papers on domination and zero forcing. In "Largest Domination Number and Smallest Independence Number of Forests with given Degree Sequence" (Gentner, Henning, Rautenbach, 2016) and "Smallest Domination Number and Largest Independence Number of Graphs and Forests with given Degree Sequence" (Gentner, Henning, Rautenbach, 2017) we examine best possible bounds for the domination number, based on the degree sequence of a graph. In some cases these bounds coincide with the Slater number of the sequence, which is a simple lower bound for the domination number. In "Some Comments on the Slater number" (Gentner, Rautenbach, 2017), we explore some more results involving the Slater number. Particularly, we determine graph classes for which the domination number is bounded from above by a term that is linear in the Slater number. In "Extremal values and bounds for the zero forcing number" (Gentner, Penso, Souza, Rautenbach, 2016) we prove two conjectures on the zero forcing number. One of these conjectures is a special case of another one, which we partially prove in "Some Bounds on the Zero Forcing Number of a Graph" (Gentner, Rautenbach, 2016). Additionally, we establish further bounds for the zero forcing number using different techniques. Apart from these papers, we explain the original motivation for the zero forcing problem and its connection to domination. While the origins of zero forcing lie in algebraic graph theory and physics, a lot of other applications can be found. One of these is the power domination problem, which also establishes the connection between domination and zero forcing