A mixed graph is a that can be obtained from simple undirected by replacing some of the edges arcs in precisely one two possible directions. The Hermitian adjacency matrix $G$ order $n$ $n \times n$ $H(G)=(h_{ij})$, where $h_{ij}=-h_{ji}= \boldsymbol{\mathrm{i}}$ (with $\boldsymbol{\mathrm{i}} =\sqrt{-1})$ if there exists an arc $v_i$ to $v_j$ (but no $v_i$), $h_{ij}=h_{ji}=1$ edge (and arcs) b...

A mixed s-stack q-queue layout of a graph consists linear order its vertices and partition edges into s stacks q queues, such that no two in the same stack cross queue nest. In 1992, Heath Rosenberg conjectured every planar admits 1-stack 1-queue layout. Recently, Pupyrev disproved this by demonstrating partial 3-tree does not admit note, we strengthen Pupyrev's result showing conjecture hold e...

Finding shortest paths in mixed graphs is NP-hard even when the weighting is conservative. We give an FPT algorithm for this problem, where the parameter k is the number of “negative trees”. We may always assume that the undirected edges have negative weights, and in a conservative weighting they form some trees – these trees are called the negative trees. Our algorithm extends to mixed graphs ...

The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound) are extremely rare, but much activity is focussed on finding new examples of graphs or families of graph with orders approaching the bound as closely as possible. There has been recent interest ...

A main question in graphical models and causal inference is whether, given a probability distribution P (which is usually an underlying distribution of data), there is a graph (or graphs) to which P is faithful. The main goal of this paper is to provide a theoretical answer to this problem. We work with general independence models, which contain probabilistic independence models as a special ca...