Christian Löwenstein

Remarks about disjoint dominating sets

We prove several structural and hardness results concerning pairs of disjoint sets in graphs which are dominating or independent and dominating.

Partitioning a graph into a dominating set, a total dominating set, and something else

A recent result of Henning and Southey (A note on graphs with disjoint dominating and total dominating set, {\it Ars Comb.} {\bf 89} (2008), 159--162) implies that every connected graph of minimum degree at least three has a dominating set

Disjoint dominating and total dominating sets in graphs

D

Generalized Power Domination in Regular Graphs

and a total dominating set

Independence, odd girth, and average degree

T

Domination Game: Extremal Families for the 3/5-Conjecture for Forests

which are disjoint. We show that the Petersen graph is the only such graph for which

On spanning tree congestion

D\cup T

Locating-dominating sets in twin-free graphs

necessarily contains all vertices of the graph.

An independent dominating set in the complement of a minimum dominating set of a tree

It has been shown [M.A. Henning, J. Southey, A note on graphs with disjoint dominating and total dominating sets, Ars Combin. 89 (2008) 159-162] that every connected graph with minimum degree at least two that is not a cycle on five vertices has a dominating set D and a total dominating set T which are disjoint. We characterize such graphs for which D@?T necessarily contains all vertices of the graph and that have no induced cycle on five vertices.

On Hamiltonian paths in distance graphs

In this paper, we continue the study of power domination in graphs (see [T. W. Haynes et al., SIAM J. Discrete Math., 15 (2002), pp. 519--529; P. Dorbec et al., SIAM J. Discrete Math., 22 (2008), pp. 554--567; A. Aazami et al., SIAM J. Discrete Math., 23 (2009), pp. 1382--1399]). Power domination in graphs was birthed from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A set of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set following a set of rules (according to Kirschoff laws) for power system monitoring. The minimum cardinality of a power dominating set of a graph is its power domination number. We show that the power domination of a connected cubic graph on