Double covers and logics of graphs
Double covers and logics of graphs. II.
Double crystals of binary and integral matrices.
Double domination critical and stable graphs upon vertex removal
In a graph a vertex is said to dominate itself and all its neighbors. A double dominating set of a graph G is a subset of vertices that dominates every vertex of G at least twice. The double domination number of G, denoted , is the minimum cardinality among all double dominating sets of G. We consider the effects of vertex removal on the double domination number of a graph. A graph G is -vertex critical graph (-vertex stable graph, respectively) if the removal of any vertex different from a support...
Double geodetic number of a graph
For a connected graph G of order n, a set S of vertices is called a double geodetic set of G if for each pair of vertices x,y in G there exist vertices u,v ∈ S such that x,y ∈ I[u,v]. The double geodetic number dg(G) is the minimum cardinality of a double geodetic set. Any double godetic of cardinality dg(G) is called dg-set of G. The double geodetic numbers of certain standard graphs are obtained. It is shown that for positive integers r,d such that r < d ≤ 2r and 3 ≤ a ≤ b there exists a connected...
Double Resolvability of Some Complete 3-Designs.
Double Schubert polynomials and degeneracy loci for the classical groups
We propose a theory of double Schubert polynomials for the Lie types , , which naturally extends the family of Lascoux and Schützenberger in type . These polynomials satisfy positivity, orthogonality and stability properties, and represent the classes of Schubert varieties and degeneracy loci of vector bundles. When is a maximal Grassmannian element of the Weyl group, can be expressed in terms of Schur-type determinants and Pfaffians, in analogy with the type formula of Kempf and Laksov....
Double-critical graphs and complete minors.
Doubly connected domination subdivision numbers of graphs
Doubly Primitive Vertex Stabilisers in Graphs.
Doubly stochastic matrices and the Bruhat order
The Bruhat order is defined in terms of an interchange operation on the set of permutation matrices of order which corresponds to the transposition of a pair of elements in a permutation. We introduce an extension of this partial order, which we call the stochastic Bruhat order, for the larger class of doubly stochastic matrices (convex hull of permutation matrices). An alternative description of this partial order is given. We define a class of special faces of induced by permutation matrices,...
Downhill Domination in Graphs
A path π = (v1, v2, . . . , vk+1) in a graph G = (V,E) is a downhill path if for every i, 1 ≤ i ≤ k, deg(vi) ≥ deg(vi+1), where deg(vi) denotes the degree of vertex vi ∈ V. The downhill domination number equals the minimum cardinality of a set S ⊆ V having the property that every vertex v ∈ V lies on a downhill path originating from some vertex in S. We investigate downhill domination numbers of graphs and give upper bounds. In particular, we show that the downhill domination number of a graph is...
Drawing 3-polytopes with good vertex resolution.
Drawing a graph in a hypercube.
Drawing clustered graphs on an orthogonal grid.
Drawing graphs on two and three lines.
Drawing graphs within graphs.
Drawing planar graphs with large vertices and thick edges.
Dual approach to edge distance between graphs
Dual codes of projective planes of order 25.