We present an experimental work and a concept based on complete, edge-colored graphs which goes towards the development of a non-growing scale-free process which would lack the disadvantages of the processes known so far i.e. a process that would i) stay in the class of simple graphs, ii) reach a scale-free degree distribution independently of the initial conditions, iii) have a constant number...

A graph is apex if there is a vertex whose deletion makes the graph planar, and doublecross if it can be drawn in the plane with only two crossings, both incident with the infinite region in the natural sense. In 1966, Tutte [9] conjectured that every two-edge-connected cubic graph with no Petersen graph minor is three-edge-colourable. With Neil Robertson, two of us showed that this is true in ...

In the path minima problem on a tree, each edge is assigned a weight and a query asks for the edge with minimum weight on a path between two nodes. For the dynamic version of the problem, where the edge weights can be updated, we give data structures that achieve optimal query time in the comparison and the RAM models. These structures also support inserting a node on an edge, inserting a leaf,...

A graph is diameter two edge-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter two edge-critical graph on n vertices is at most ⌊ n2 4 ⌋ and the extremal graph is the complete bipartite graph Kb 2 c,d 2 e. In the series papers [7–9], the Murty-Simon Conjecture stated by Haynes et al. is not the...

Calculating the most reliable maximum flow (MRMF) from edge cache node to requesting can provide an important reference for selecting best in a content delivery network (CDN). However, SDBA, as current state-of-the-art MRMF algorithm, is too complex meet real-time computing needs. This paper proposes set of algorithms: NWCD (Negative Weight Community Deletion), SCPDAT (Single-Cycle Preference D...

‎let $g=(v(g),e(g))$ be a simple connected graph with vertex set $v(g)$ and edge‎ ‎set $e(g)$‎. ‎the (first) edge-hyper wiener index of the graph $g$ is defined as‎: ‎$$ww_{e}(g)=sum_{{f,g}subseteq e(g)}(d_{e}(f,g|g)+d_{e}^{2}(f,g|g))=frac{1}{2}sum_{fin e(g)}(d_{e}(f|g)+d^{2}_{e}(f|g)),$$‎ ‎where $d_{e}(f,g|g)$ denotes the distance between the edges $f=xy$ and $g=uv$ in $e(g)$ and $d_{e}(f|g)=s...

Introduction: Gaseous air pollutants can cause oxidative stress, which can lead to lung damage by inducing inflammation. Polymorphisms in the glutathione S-transferase (GST) gene are involved in the pathogenesis of many diseases, including lung disease. Two glutathione S-transferase Mu 1 (GSTM1) and glutathione S-transferase theta 1 (GSTT1) genes belong to this family, in which deletions occur ...

We study the effect of subtle changes on the evolution in the scale-free (SF) networks. Three extended models are evolved based on competition and inner anti-preferential deletion in growth and preferential attachment processes. By nonlinear and dynamic controlling on randomness and determinacy, three models can self-organize into scale-free networks, and diverse scaling exponents appear. Moreo...

Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also in...

Luke Mathieson – The Parameterized Complexity of Degree Constrained Editing Problems This thesis examines degree constrained editing problems within the framework of parameterized complexity. A degree constrained editing problem takes as input a graph and a set of constraints and asks whether the graph can be altered in at most k editing steps such that the degrees of the remaining vertices are...