The domination multisubdivision number of a nonempty graph G was defined in [3] as the minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. Similarly we define the total domination multisubdivision number msdγt(G) of a graph G and we show that for any connected graph G of order at least two, msdγt(G) ≤ 3. We show that...

A graph G is domination dot-critical, or just dot-critical, if contracting any edge decreases the domination number. It is totally dot-critical if identifying any twovertices decreases the domination number. In this paper, we study an open question concerning of the diameter of a domination dot-critical graph G. © 2008 Elsevier B.V. All rights reserved.

In this paper, we consider the problem of reducing semitotal domination number a given graph by contracting k edges, for some fixed k≥1. We show that can always be done with at most 3 edge contractions and further characterise those graphs requiring 1, 2 or contractions, respectively, to decrease their number. then study complexity k=1 obtain in particular complete dichotomy monogenic classes.

For any edge of an isolate free graph , , 〈 〉 is the subgraph induced by the vertices adjacent to u and v in G. We say that an edge x, edominates an edge y if ∈ 〈 〉 . A set ⊆ is an Edge-Edge Dominating Set (EED-set) if every edge in is e-dominated by an edge in L. The edgeedge domination number is the cardinality of a minimum EED-set. We find the relation ship between the new parameter and some...

The open neighborhood NG(e) of an edge e in a graph G is the set consisting of all edges having a common end-vertex with e and its closed neighborhood is NG[e] = NG(e) ∪ {e}. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If ∑x∈NG[e] f(x) ≥ 1 for at least a half of the edges e ∈ E(G), then f is called a signed edge majority dominating function of G. The minimum of the val...

For any $k in mathbb{N}$, the $k$-subdivision of graph $G$ is a simple graph $G^{frac{1}{k}}$, which is constructed by replacing each edge of $G$ with a path of length $k$. In [Moharram N. Iradmusa, On colorings of graph fractional powers, Discrete Math., (310) 2010, No. 10-11, 1551-1556] the $m$th power of the $n$-subdivision of $G$ has been introduced as a fractional power of $G$, denoted by ...

Let D be the minimum dominating set of intuitionistic fuzzy graph G. The minimum intuitionistic fuzzy cardinality of all edge dominating set of intuitionistic fuzzy graph G is known as edge domination number and it is denoted by γe(G). In this Paper, we initiate some definitions onedge dominating set concerning intuitionistic fuzzy sets. Further, we investigate some results onedge domination nu...