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1

Connected domination polynomial of graphs

Mojdeh D. A., Emadi A. S.

Fasciculi Mathematici

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2018

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Nr 60

103--121

EN

Let G be a simple graph of order n. The connected domination polynomial of G is the polynomial Dc (G, x) = Σ |V(G)| i=γc(G) dc (G, i)xi, where dc (G, i) is the number of connected dominating sets of G of size i and γc (G) is the connected domination number of G. In this paper we study Dc (G, x) of any graph. We classify many families of graphs by studying their connected domination polynomial.

2

On the inverse signed total domination number in graphs

Mojdeh D. A., Samadi B.

Opuscula Mathematica

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2017

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Vol. 37, no. 3

447--456

EN

In this paper, we study the inverse signed total domination number in graphs and present new sharp lower and upper bounds on this parameter. For example by making use of the classic theorem of Turán (1941), we present a sharp upper bound on Kr+1-free graphs for r ≥ 2. Also, we bound this parameter for a tree from below in terms of its order and the number of leaves and characterize all trees attaining this bound.

3

Total limited packing in graphs

Hosseini Moghaddam S. M., Mojdeh D. A., Samadi B.

Fasciculi Mathematici

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2016

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Nr 56

121--127

EN

We define a k-total limited packing number in a graph, which generalizes the concept of open packing number in graphs, and give several bounds on it. These bounds involve many well known parameters of graphs. Also, we establish a connection among the concepts of tuple domination, tuple total domination and total limited packing that implies some results.

4

On the diameter of dot-critical graphs

Mojdeh D. A., Mirzamani S.

Opuscula Mathematica

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2009

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Vol. 29, no. 2

165-175

EN

A graph G is k-dot-critical (totally k-dot-critical) if G is dot-critical (totally dot-critical) and the domination number is k. In the paper [T. Burtona, D. P. Sumner, Domination dot-critical graphs, Discrete Math, 306(2006), 11-18] the following question is posed: What are the best bounds for the diameter of a k-dot-critical graph and a totally k-dot-critical graph G with no critical vertices for k ≥ 4? We find the best bound for the diameter of a k-dot-critical graph, where k ∈ {4, 5, 6} and we give a family of k-dot-critical graphs (with no critical vertices) with sharp diameter 2k - 3 for even k ≥ 4.

5

Strong geodomination in graphs

Rad N. J., Mojdeh D. A.

Opuscula Mathematica

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2008

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Vol. 28, no. 3

279-285

EN

A pair x, y of vertices in a nontrivial connected graph G is said to geodominate a vertex v of G if either v ∈ {x, y} or v lies in an x - y geodesic of G. A set S of vertices of G is a geodominating set if every vertex of G is geodominated by some pair of vertices of S. In this paper we study strong geodomination in a graph G.

6

k-Perfect geodominating sets in graphs

Mojdeh D. A., Rad N. J.

Opuscula Mathematica

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2007

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Vol. 27, no. 1

51-57

EN

A perfect geodominating set in a graph G is a geodominating set S such that any vertex v ∈ V(G)\S is geodominated by exactly one pair of vertices of S. A k-perfect geodominating set is a geodominating set S such that any vertex v ∈ V(G)\S is geodominated by exactly one pair x, y of vertices of S with d(x, y) = k. We study perfect and k-perfect geodomination numbers of a graph G.