We answer a question of Jockusch by showing that the measure of the Turing degrees which satisfy the cupping property is 0. In fact, every 2-random degree has a strong minimal cover, and so fails to satisfy the cupping

An independent broadcast on a connected graph $G$is a function $f:V(G)to mathbb{N}_0$such that, for every vertex $x$ of $G$, the value $f(x)$ is at most the eccentricity of $x$ in $G$,and $f(x)>0$ implies that $f(y)=0$ for every vertex $y$ of $G$ within distance at most $f(x)$ from $x$.The broadcast independence number $alpha_b(G)$ of $G$is the largest weight $sumlimits_{xin V(G)}f(x)$of an ind...

Several new graph theoretic problems, which arise naturally from existing coloring algorithms, are de ned. The complementary High Degree Vertex Removal Problem and Low Degree Vertex Removal Problem are both shown to be NP-complete. The Low Degree Subgraph Problem is de ned and shown to be NP-complete, whereas, the \complementary" problem for high degree subgraphs was previously shown to be P-co...

For a graph G, the irregularity and total irregularity of G are defined as irr(G)=∑_(uv∈E(G))〖|d_G (u)-d_G (v)|〗 and irr_t (G)=1/2 ∑_(u,v∈V(G))〖|d_G (u)-d_G (v)|〗, respectively, where d_G (u) is the degree of vertex u. In this paper, we characterize all ‎connected Eulerian graphs with the second minimum irregularity, the second and third minimum total irregularity value, respectively.

Fellows, Guo, Moser and Niedermeier [JCSS2011] proved a generalization of Nemhauser and Trotter’s theorem, which applies to Bounded-Degree Vertex Deletion (for a fixed integer d ≥ 0, to delete k vertices of the input graph to make the maximum degree of it ≤ d) and gets a linear-vertex kernel for d = 0 and 1, and a superlinear-vertex kernel for each d ≥ 2. It is still left as an open problem whe...