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2 Discussiones Mathematicae Graph Theory

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Some Variations of Perfect Graphs

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Dettlaff M. , Lemańska M. , Semanišin G. , Zuazua R.

Discussiones Mathematicae Graph Theory

2016

tom 36

nr 3

661-668

We consider (ψk−γk−1)-perfect graphs, i.e., graphs G for which ψk(H) = γk−1(H) for any induced subgraph H of G, where ψk and γk−1 are the k-path vertex cover number and the distance (k − 1)-domination number, respectively. We study (ψk−γk−1)-perfect paths, cycles and complete graphs for k ≥ 2. Moreover, we provide a complete characterisation of (ψ2 − γ1)- perfect graphs describing the set of its forbidden induced subgraphs and providing the explicit characterisation of the structure of graphs belonging to this family.

Total Domination Multisubdivision Number of a Graph

100%

Avella-Alaminos D. , Dettlaff M. , Lemańska M. , Zuazua R.

tom 35

nr 2

315-327

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 for trees the total domination multisubdi- vision number is equal to the known total domination subdivision number. We also determine the total domination multisubdivision number for some classes of graphs and characterize trees T with msdγt (T) = 1.