True trees are dense

True Trees Are Dense

We show that any compact, connected set K in the plane can be approximated by the critical points of a polynomial with two critical values. Equivalently, K can be approximated in the Hausdorff metric by a true tree in the sense of Grothendieck’s dessins d’enfants. Mathematics Subject Classification 30C62

Spanning Trees In Dense Graphs

1.1. Notation and definitions We will sometimes use + and Σ to denote disjoint unions of sets. V (G) and E(G) denote the vertex set and the edge set of the graph G, and we write v(G) = |V (G)| (order of G) and e(G) = |E(G)| (size of G). (A,B) or (A,B, E) denote a bipartite graph G = (V , E), where V = A+B, and E ⊂ A×B. In general, given any graph G and two disjoint subsets A,B of V (G), the pai...

Some New Results for k - Dense Trees

Degree sums and dense spanning trees

Finding dense spanning trees (DST) in unweighted graphs is a variation of the well studied minimum spanning tree problem (MST). We utilize established mathematical properties of extremal structures with the minimum sum of distances between vertices to formulate some general conditions on the sum of vertex degrees. We analyze the performance of various combinations of these degree sum conditions...

Are software engineers true engineers?

Are software engineers engineers? Or more exactly: Can software engineers legitimately be called engineers? This is an ambiguous question. Its interpretation depends on the situation in which it is asked. It might be asked in a situation in which practice-oriented engineering disciplines have more chances to be sponsored publicly: In this case it would be the question if software engineering sh...