We disprove a conjecture by Skupień that every tree of order n has at most 2 minimal dominating sets. We construct a family of trees of both parities of the order for which the number of minimal dominating sets exceeds 1.4167. We also provide an algorithm for listing all minimal dominating sets of a tree in time O(1.4656). This implies that every tree has at most 1.4656 minimal dominating sets.

In this paper we consider 1-movable dominating sets, motivated by the use of sensors employed to detect certain events in networks, where the sensors have a limited ability to react under changing conditions in the network. A 1-movable dominating set is a dominating set S ⊆ V (G) such that for every v ∈ S, either S − {v} is a dominating set, or there exists a vertex u ∈ (V (G) − S) ∩ N(v) such ...

We first consider some problems related to the maximum number of dominating (or strong dominating) sets in a regular graph. Our techniques, centered around Shearer’s entropy lemma, extend to a reasonably broad class of graph parameters enumerating vertex colorings that satisfy conditions on the multiset of colors appearing in neighborhoods (either open or closed). Dominating sets and strong dom...

A graph is said to be well-dominated if all its minimal dominating sets are of the same size. In this work, we introduce the notion of an irreducible dominating set, a variant of dominating set generalizing both minimal dominating sets and minimal total dominating sets. Based on this notion, we characterize the family of minimal dominating sets in a lexicographic product of two graphs and deriv...

We show that interval graphs on n vertices have at most 3 ≈ 1.4422 minimal dominating sets, and that these can be enumerated in time O∗(3n/3). As there are examples of interval graphs that actually have 3 minimal dominating sets, our bound is tight. We show that the same upper bound holds also for trees, i.e. trees on n vertices have at most 3 ≈ 1.4422 minimal dominating sets. The previous best...

A set S of vertices in a graph G=(V,E) is a dominating set ofG if every vertex of V-S is adjacent to some vertex of S.For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from somevertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongestdominating set as well a...

We consider the problem of incrementally computing a minimal dominating set of a directed graph after the insertion or deletion of a set of arcs. Earlier results have either focused on the study of the properties that minimum (not minimal) dominating sets preserved or lacked to investigate which update affects a minimal dominating set and in what ways. In this paper, we first show how to increm...

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.

Motivated by an application to unstructured multigrid calculations, we consider the problem of asymptotically minimizing the size of dominating sets in triangulated planar graphs. Speci cally, we wish to nd the smallest such that, for n su ciently large, every n-vertex planar graph contains a dominating set of size at most n. We prove that 1 4 13 , and we conjecture that = 14 . For triangulated...

In this paper we study the size of generalised dominating sets in two graph processes which are widely used to model aspects of the world-wide web. On the one hand, we show that graphs generated this way have fairly large dominating sets (i.e. linear in the size of the graph). On the other hand, we present efficient strategies to construct small dominating sets. The algorithmic results represen...