A set S of vertices in a graph G(V,E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. The total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. Total domination subdivision number denoted by sdγt is the minimum number of edges that must be subdivided to increase the total domination number. Here we...

A graph G is said to have property Pm if it contains no subdivision of Km+1 and no subdivision of Kdm=2e+1;bm=2c+1. Chartrand et al. (J. Combin Theory 10 (1971) 12–41) (see also Problem 6.3 in Jensen and Toft (Graph Coloring Problems, Wiley, New York, 1995) conjectured that the set of vertices (respectively, edges) of any graph with property Pm can be partitioned into m−n+1 subsets such that ea...

We analyze the space Sr,r ′ m,m (T ) of bivariate functions that are piecewise polynomial of bidegree (m,m) and class C ′ over the planar T-subdivision T . We give a new formula for the dimension of this space by exploiting homological techniques. We relate this dimension to the number of nodes on the maximal interior segments of the subdivision, give combinatorial lower and upper bounds on the...