Remarks about disjoint dominating sets

We solve a number of problems posed by Hedetniemi, Hedetniemi, Laskar, Markus, and Slater concerning pairs of disjoint sets in graphs which are dominating or independent and dominating

Disjoint Dominating Sets with a Perfect Matching

In this paper, we consider dominating sets $D$ and $D'$ such that $D$ and $D'$ are disjoint and there exists a perfect matching between them. Let $DD_{\textrm{m}}(G)$ denote the cardinality of smallest such sets $D, D'$ in $G$ (provided they exist, otherwise $DD_{\textrm{m}}(G) = \infty$). This concept was introduced in [Klostermeyer et al., Theory and Application of Graphs, 2017] in the context of studying a certain graph protection problem. We characterize the trees $T$ for which $DD_{\textrm{m}}(T)$ equals a certain graph protection parameter and for which $DD_{\textrm{m}}(T) = \alpha(T)$, where $\alpha(G)$ is the independence number of $G$. We also further study this parameter in graph products, e.g., by giving bounds for grid graphs, and in graphs of small independence number

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, Ars Comb. 89 (2008), 159-162) implies that every connected graph of minimum degree at least three has a dominating set D and a total dominating set T which are disjoint. We show that the Petersen graph is the only such graph for which D∪T necessarily contains all vertices of the graph

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 $D$ and a total dominating set $T$ which are disjoint. We show that the Petersen graph is the only such graph for which $D\cup T$ necessarily contains all vertices of the graph

On dominating graph of graphs, median graphs and partial cubes, and graphs in which complement of every minimal dominating set is minimal dominating

The dominating graph of a graph G is a graph whose vertices correspond to the dominating sets of G and two vertices are adjacent whenever their corresponding dominating sets differ in exactly one vertex. Studying properties of dominating graph has become an increasingly interesting subject in domination theory. On the other hand, median graphs and partial cubes are two fundamental graph classes in graph theory. In this paper, we make some new connections between domination theory and the theory of median graphs and partial cubes. As the main result, we show that the following conditions are equivalent for every graph $G \not \simeq C_4$ with no isolated vertex, and in particular, that the simple third condition completely characterizes first two ones in which three concepts of dominating graphs, median graphs and complement of minimal dominating sets get related: - The dominating graph of G is a median graph, - The complement of every minimal dominating set of G is a minimal dominating set, - Every vertex of G is either of degree 1 or adjacent to a vertex of degree 1. As another result, we prove that the dominating graph of every graph is a partial cube and also give some examples to show that not all partial cubes or median graphs are isomorphic to the dominating graph of a graph. The above-mentioned results, as another highlight of the paper, provide novel infinite sources of examples of median graphs and partial cubes