# Total outer-connected domination in trees

@article{Cyman2010TotalOD, title={Total outer-connected domination in trees}, author={Joanna Cyman}, journal={Discuss. Math. Graph Theory}, year={2010}, volume={30}, pages={377-383} }

Let G = (V, E) be a graph. Set D ⊆ V (G) is a total outerconnected dominating set of G if D is a total dominating set in G and G[V (G)−D] is connected. The total outer-connected domination number of G, denoted by γtc(G), is the smallest cardinality of a total outer-connected dominating set of G. We show that if T is a tree of order n, then γtc(T ) ≥ d 2n 3 e. Moreover, we constructively characterize the family of extremal trees T of order n achieving this lower bound.

## 15 Citations

### A note on the total outer-connected domination number of a tree

- Mathematics
- 2010

Let G = (V, E) be a graph. A set D ⊆ V is a total outer-connected dominating set of G if D is dominating and G[V − D] is connected. The total outer-connected domination number of G, denoted γtc(G),…

### On the total outer-connected domination in graphs

- MathematicsJ. Comb. Optim.
- 2014

For graphs of order n, diameter 2 and minimum degree at least 3, it is shown that γtoc (G) is the minimum size of a total outer-connected dominating set and that $\gamma_{toc}(G)le 2n-2}{3}$ and the extremal graphs are determined.

### Total outer-Connected domination subdivision numbers in graphs

- MathematicsDiscret. Math. Algorithms Appl.
- 2013

The existence of sdγtoc(G) is proved for every connected graph G of order at least 3 and upper bounds on it are given in some classes of graphs.

### Generalization of the total outer-connected domination in graphs

- MathematicsRAIRO Oper. Res.
- 2016

The total outer- k -connected component domination number of G, denoted by γ k tc ( G ) , is the minimum cardinality of a total inner-connected component dominating set of G.

### Complexity of total outer-connected domination problem in graphs

- MathematicsDiscret. Appl. Math.
- 2016

### On the Secure Total Domination Number of Graphs

- MathematicsSymmetry
- 2019

In this article, new relationships between the secure total domination number and other graph parameters are obtained: namely the independence number, the matchingnumber and other domination parameters.

### Outer-convex Domination in the Composition and Cartesian Product of Graphs

- Mathematics
- 2019

Let G be a connected simple graph. A set S of vertices of a graph G is an outer-convex dominating set if every vertex not in S is adjacent to some vertex in S and V (G) \ S is a convex set. In this…

### Some Classical Problems on K1,3-free Split Graphs.

- Mathematics2020 IEEE 4th Conference on Information & Communication Technology (CICT)
- 2020

A set D$\subseteq$V of a graph G is called a total outer-connected dominating set of G if D is a total dominating set of G and G[V$\backslash$D] is connected. The Minimum Total Outer-connected…

### Inequality of Nordhaus-Gaddum type for total outer-connected domination in graphs

- Mathematics
- 2011

AbstractA set S of vertices in a graph G = (V, E) without isolated vertices is a total outer-connected dominating set (TCDS) of G if S is a total dominating set of G and G[V − S] is connected. The…

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