A set of vertices is a dominating set in a graph if every vertex not in the dominating set is adjacent to one or more vertices in the dominating set. A dominating clique is a dominating set that induces a complete subgraph. Forbidden subgraph conditions sufficient to imply the existence of a dominating clique are given. For certain classes of graphs, a polynomial algorithm is given for finding ...

In 2008, Favaron and Henning proved that if G is a connected claw-free cubic graph of order n ≥ 10, then the total domination number γt(G) of G is at most 5 11 n, and they conjectured that in fact γt(G) is at most 4 9 n (see [O. Favaron and M.A. Henning, Discrete Math. 308 (2008), 3491–3507] and [M.A. Henning, Discrete Math. 309 (2009), 32–63]). In this paper, in a first step, we prove this con...

Let G be a graph. A total dominating set of G is a set S of vertices of G such that every vertex is adjacent to at least one vertex in S. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of G. In this paper we would like to characterize the cubic graphs with total domatic number at least two.

We investigate the structure of the degrees of provability, which measure the proof-theoretic strength of statements asserting the totality of given computable functions. The degrees of provability can also be seen as an extension of the investigation of relative consistency statements for firstorder arithmetic (which can be viewed as Π1-statements, whereas statements of totality of computable ...

We consider two general frameworks for multiple domination, which are called 〈r, s〉-domination and parametric domination. They generalise and unify {k}-domination, k-domination, total k-domination and k-tuple domination. In this paper, known upper bounds for the classical domination are generalised for the 〈r, s〉-domination and parametric domination numbers. These generalisations are based on t...

Given a graph G = (V,E), a mixed dominating set MD of G is defined to be a subset of V ∪ E such that every element in {(V ∪E)\MD} is either adjacent or incident to an element of MD. The mixed dominating set problem is to find a mixed dominating set with minimum cardinality. This problem is NP-hard. In this paper, we prove that this problem is MAX SNP-hard.

Given a graph G, we say that a subset D of the vertex set V is a dominating set if it is near all the vertices, in that every vertex outside of D is adjacent to a vertex in D. A domatic k-partition of G is a partition of V into k dominating sets. In this paper, we will consider issues of computability related to domatic partitions of computable graphs. Our investigation will center on answering...

Motivated by a question of Krzysztof Oleszkiewicz we study a notion of weak tail domination of random vectors. We show that if the dominating random variable is sufficiently regular then weak tail domination implies strong tail domination. In particular positive answer to Oleszkiewicz question would follow from the so-called Bernoulli conjecture. We also prove that any unconditional logarithmic...

In 1996, Reed proved that the domination number, γ(G), of every n-vertex graph G with minimum degree at least 3 is at most 3n/8 and conjectured that γ(H) ≤ dn/3e for every connected 3-regular (cubic) n-vertex graph H. In [1] this conjecture was disproved by presenting a connected cubic graph G on 60 vertices with γ(G) = 21 and a sequence {Gk} ∞ k=1 of connected cubic graphs with limk→∞ γ(Gk) |V...

The node-weighted Steiner tree problem is a variation of classical Steiner minimum tree problem. Given a graph G = (V,E) with node weight function C : V → R and a subset X of V , the node-weighted Steiner tree problem is to find a Steiner tree for the set X such that its total weight is minimum. In this paper, we study this problem in unit disk graphs and present a (1+ε)-approximation algorithm...