1. Outer-connected domination numbers of block graphs 杜國豪 指導教授：郭大衛教授 國立東華大學 應用數學系碩士班

2. Outline:  Introduction  Main result Full k-ary tree Block graph  Reference

3. Definition:  For a graph a set is a dominating set if.  A dominating set is an outer-connected dominating set(OCD set) if the subgraph induced by is connected. Example:

4. Definition:  For a graph a set is a dominating set if.  A dominating set is an outer-connected dominating set(OCD set) if the subgraph induced by is connected. Example:

5. Definition:  A full -ary tree with height denoted is a k-ary tree with all leaves are at same level.

6. Proposition 1:  If is a tree and is an outer-connected dominating set of, then either or every leaf of belongs to Lemma 2:  If is a cut-vertex of and are the components of then for every outer- connected dominating set of which contains there exists such that

7. Theorem 3: For all,

8. Theorem 4: For all

9. Definition:  A block of a graph is a maximal -connected subgraph of  A block graph is a graph which every block is a complete graph.  The block-cut-vertex tree of a graph is a bipartite graph in which one partite set consists of the cut-vertices of, and the other has a vertex for each block of And adjacent to, if containing in

10. Example:

14. Red: cut-vertex Blue: block

15. Example:

17. Algorithm for block graphs:  

18.          

19.                

20. Initial values:  Time complexity:  Each vertex uses a constant time for computing its parameters, the time complexity of this algorithm is

21. Example 1:

28. Example 2:

29. Red: cut-vertex Blue: block

30. Example 2:

32. Example 3:

33. Red: cut-vertex Blue: block Example 3:

36. Reference:  Akhbari, R. Hasni, O. Favaron, H. Karami and S. M. Sheikholeslami, "On the outer-connected domination in graphs," J. Combin. Optimi. DOI 10.1007/s10878-011-9427-x (2011).  J. Cyman, The outer-connected domination number of a graph, Australas. J. Combin., 38 (2007), 35-46.  H. Jiang and E. Shan, Outer-connected domination number in graph, Utilitas Math., 81 (2010), 265-274.

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