MacGillivray and Seyffarth (J. Graph Theory 22 (1996), 213–229) proved that planar graphs of diameter three have domination number at most ten. Recently it was shown (J. Graph Theory 40 (2002), 1–25) that a planar graph of diameter three and of radius two has domination number at most six while every sufficiently large planar graph of diameter three has domination number at most seven. In this ...

We consider four different types of multiple domination and provide new improved upper bounds for the kand k-tuple domination numbers. They generalise two classical bounds for the domination number and are better than a number of known upper bounds for these two multiple domination parameters. Also, we explicitly present and systematize randomized algorithms for finding multiple dominating sets...

An exact lower bound for the domination number and the total domination number of the direct product of finitely many complete graphs is given: (×i=1Kni) ≥ t + 1, t ≥ 3. Sharpness is established in the case when the factors are large enough in comparison to the number of factors. The main result gives a lower bound for the domination (and the total domination) number of the direct product of tw...

Domination is a relation between general operations defined on a poset. The old open problem is whether domination is transitive on the set of all t-norms. In this paper we contribute partially by inspection of domination in the family of Frank and Hamacher tnorms. We show that between two different t-norms from the same family, the domination occurs iff at least one of the t-norms involved is ...

let $d=(v,a)$ be a finite simple directed graph. a function$f:vlongrightarrow {-1,0,1}$ is called a twin minus dominatingfunction (tmdf) if $f(n^-[v])ge 1$ and $f(n^+[v])ge 1$ for eachvertex $vin v$. the twin minus domination number of $d$ is$gamma_{-}^*(d)=min{w(f)mid f mbox{ is a tmdf of } d}$. inthis paper, we initiate the study of twin minus domination numbersin digraphs and present some lo...

A set S of vertices of a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by i(G), is the minimum cardinality of an independent dominating set of G. In this paper, some new classes of graphs with equal domination and independent domination numbers are presented and exa...

In this note, we prove several lower bounds on the domination number of simple connected graphs. Among these are the following: the domination number is at least two-thirds of the radius of the graph, three times the domination number is at least two more than the number of cut-vertices in the graph, and the domination number of a tree is at least as large as the minimum order of a maximal matc...

Numerous studies concerning the number and disposition of chordless paths on four vertices in graphs proved that there is a strong relation between these paths and the perfectness of graphs. In this paper, we further investigate this relation by introducing the notion of P4-domination and formulating two conjectures, which are used both to find an equivalent version of the Odd Pair Conjecture a...

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. We call these graphs γt-critical. In this paper, we determine the domination and the total domination number in the Circulant graphs Cn〈1, 3〉, and then study γ-criticality...