# Improved Networks Routing Using an Arrow-Based Description
## 2 Basic Definitions and Notations  
### Background  
It is known that any graph is represented by a set of links, which are connected to each other via nodes [7]. The number of links departing or arriving in a node is called the _degree_ of the node. A link connected to a node is said to be _incident_ to the node. A suite of links connecting two end nodes by passing through other nodes is called a _path_. A closed path, i.e., a path starting and ending in the same node is called a _cycle_. A cycle that visits every node of the graph exactly once is called a _Hamiltonian_ cycle; a graph that has at least one Hamiltonian cycle is called a _Hamiltonian graph_. A link connected to a cycle, i.e., a component of a given cycle, is said to be _incident_ to the cycle. Two cycles are _adjacent_ if they share only a single link.  
The total number of nodes in the graph is denoted as N and the number of links is denoted as L. In this paper, we use simple planar connected graphs, which have no self-looping nodes, no leaves (nodes of degree 1) and no parallel links between any two nodes.  
Algebraically, the graph's connectivity can be described by using either the nodes-links or the cycles-links incidence matrix. The nodes-links incidence matrix \(\mathbf{A}\) is an N\(\times\)L matrix (it has N rows and L columns), whose elements describe which links connect which nodes. The cycles-links incidence matrix \(\mathbf{B}\) is a C\(\times\)L matrix, where C is the number of independent cycles, and L, the number of links of the graph. In a planar graph, the values C, L, and N are such that [8] they fulfill the Euler equation C = L-N + 2.  
Labeling is used to uniquely identify different parts of the graph, such as nodes, links, and cycles. In addition, graphs representing nodes are usually undirected. However, a randomly chosen direction is used in order to, e.g., analyze network functions. Neither the chosen labeling, nor the added direction, directly influence the properties of the graph.  
In a planar graph, its cycles divide the geometric plane into an internal and an external region, and a cycle defines the border between the two regions. In this paper, we index the cycles such that the set of the cycles C is defined as \(\mathbf{C}\) = {C1, C2,..., C\({}_{c}\)} and we assume that the cycle with the highest index C\({}_{c}\) is the border cycle, and c is given by the Euler equation. Recall that a planar graph is a graph that can be drawn in a planar embedding such that no link crossings are present.  
We present the cycles description of a graph subsequently. This description can be adopted for any simple, planar, cyclically connected graph [1].  
In this paper, we will use an incidence matrix describing how the links are related to the cycles. This matrix is the cycles-links incidence matrix \(\mathbf{B_{C,L}}\) of the associated graph, which will be presented in a later paragraph. It is orthogonal to the nodes-links incidence matrix \(\mathbf{A}\). [7, 8].
We depict the result of the merger, denoted \(c_{1}|c_{2}\) in Figure 2.  
The authors of two papers [9, 10] describe an algorithm of obtaining the Hamiltonian cycles in a graph by using cycle merger. This method needs no backtracking and thus the algorithm is parallelizable.