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Outer-connected domination numbers of block graphs

Outer-connected domination numbers of block graphs

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Outer-connected domination numbers of block graphs

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  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: • Afull -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:

  11. Example:

  12. Example:

  13. Example:

  14. Red: cut-vertex Blue: block Example:

  15. Example:

  16. Example:

  17. Algorithm for block graphs:

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

  19. Example 1:

  20. Example 1:

  21. Example 1:

  22. Example 1:

  23. Example 1:

  24. Example 1:

  25. Example 1:

  26. Example 2:

  27. Red: cut-vertex Blue: block Example 2:

  28. Example 2:

  29. Example 2:

  30. Example 3:

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

  32. Example 3:

  33. Example 3:

  34. 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.

  35. THANK YOU!