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## Outer-connected domination numbers of block graphs

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**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**Outline:**• Introduction • Main result • Full k-ary tree • Block graph • Reference**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:**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:**Definition:**• Afull -ary tree with height denoted is a k-ary tree with all leaves are at same level.**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**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**Red: cut-vertex**Blue: block Example:**Initial values:**Time complexity: • Each vertex uses a constant time for computing its parameters, the time complexity of this algorithm is**Red: cut-vertex**Blue: block Example 2:**Example 3:**Red: cut-vertex Blue: block**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.