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A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd γt (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper we first prove that sdγt(G) < n -δ + 2 for every simple connected graph G of order n ≥ 3. We also classify all simple connected graphs G with sdγt (G) = n -δ+ 2,n-δ + 1, and n-δ  

It is possible to define multiresolution by reversing the process of subdivision. One approach to reverse a subdivision scheme appropriates pure numerical algebraic relations for subdivision using the interaction of diagrams (Bartels and Samavati, 2011; Samavati and Bartels, 2006). However, certain assumptions carried through the available work, two of which we wish to challenge: (1) the construction of multiresolutions for irregular meshes are reconsidered in the presence of any extraordinary vertex rather than being prepared beforehand as simple available relations and (2) the connectivity graph of the coarse mesh would have to be a subgraph of the connectivity graph of the fine mesh. 3... 

We have shown how to construct multiresolution structures for reversing subdivision rules using global least squares models (Samavati and Bartels, Computer Graphics Forum, 18(2):97-119, June 1999). As a result, semiorthogonal wavelet systems have also been generated. To construct a multiresolution surface of an arbitrary topology, however, biorthogonal wavelets are needed. In Bartels and Samavati (Journal of Computational and Applied Mathematics, 119:29-67, 2000) we introduced local least squares models for reversing subdivision rules to construct multiresolution curves and tensor product surfaces, noticing that the resulting wavelets were biorthogonal (under an induced inner product). Here,... 

A set S of vertices of a graph G = (V, E) is a dominating set if every vertex of V(G) S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G. The domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Velammal in his Ph.D. thesis [Manonmaniam Sundaranar University, Tirunelveli, 1997] showed that for any tree T of order at least 3, 1 ≤ sdγ(T) ≤ 3. Furthermore, Aram, Favaron and Sheikholeslami, recently, in their paper entitled "Trees with domination subdivision number three," gave two characterizations of... 

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S.The total domination number γ<(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sdγt(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. We show that sdγt(G) ≤ n - γt(G) + 1 for any graph G of order n ≥ 3 and that sdγt(G) < n-γt(G) except if G ≃ P3, C3, K4, P6 or C6  

A set S of vertices of a graph G = (V, E) with no isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domination numberγt (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision numbersdγt (G) is the minimum number of edges that must be subdivided in order to increase the total domination number. We consider graphs of order n ≥ 4, minimum degree δ and maximum degree Δ. We prove that if each component of G and over(G, -) has order at least 3 and G, over(G, -) ≠ C5, then γt (G) + γt (over(G, -)) ≤ frac(2 n, 3) + 2 and if each component of G and over(G, -) has order at least 2 and at least one...