Abstract For a graph G , two dominating sets D and $$D'$$ D ′ in non-negative integer k the set is said to -transform if there sequence $$D_0,\ldots ,D_\ell $$ 0 , … ℓ ...

We develop the problem of fault-tolerant dominating sets (liar’s dominating sets) in graphs. Namely, we consider a new kind of fault – a false alarm. Characterization of such fault-tolerant dominating sets in three different cases (dependent on the classification of the types of the faults) are presented.

A DD2-pair of a graph G is a pair (D,D2) of disjoint sets of vertices of G such that D is a dominating set and D2 is a 2-dominating set of G. Although there are infinitely many graphs that do not contain a DD2-pair, we show that every graph with minimum degree at least two has a DD2-pair. We provide a constructive characterization of trees that have a DD2-pair and show that K3,3 is the only con...

A graph is a homogeneous extension of a tree iff the reduction of all homogeneous sets (sometimes called modules) to single vertices gives a tree. We show that these graphs can be recognized in linear sequential and polylogarithmic parallel time using modular decomposition. As an application of some results on homogeneous sets we present a linear time algorithm computing the vertex sets of the ...

We make an attempt to understand the dominating set counting problem in graph classes from the viewpoint of polynomial-time computability. We give polynomial-time algorithms to count the number of dominating sets (and minimum dominating sets) in interval graphs and trapezoid graphs. They are based on dynamic programming. With the help of dynamic update on a binary tree, we further reduce the ti...

The maximum number of minimal dominating sets that a graph on n vertices can have is known to be at most 1.7159. This upper bound might not be tight, since no examples of graphs with 1.5705 or more minimal dominating sets are known. For several classes of graphs, we substantially improve the upper bound on the maximum number of minimal dominating sets in graphs on n vertices. In some cases, we ...

We conduct an extensive numerical analysis of the effects of network damage on dominating sets that occur at either randomly selected nodes or at the highest-degree nodes, in synthetic scale-free and real complex networks. We focus on the fraction of nodes that remain dominated after the damage has occurred, defined as stability. We find that minimum dominating sets (MDS), despite being optimal...

Given a graph class G, it is natural to ask (i) whether a given graph G has a connected or a total dominating set whose induced subgraph is a member of G and (ii) whether G has such a set of cardinality at most a given threshold. We give sufficient conditions on G under which these two problems are NP-hard. This condition is fulfilled in a wide variety of classes of graphs.

In this paper we deal with different type of dominating sets in de Bruijn graphs and we prove a conjecture on perfect dominating sets.