# Graph Theory Approach to the Vulnerability of Transportation Networks
## 3 Results
### Structural Transportation Network Vulnerability
### Customer Vulnerability

When we take into account the concept of domination subdivision number, the parameter of customer vulnerability can be defined. This parameter allows determining the sensitivity of transportation or logistics network according to an increase in the number of clients relative to service (located in the node). In other words, it can also determine how much the transport network is resistant to changes in the number of customers. This parameter is an essential characteristic, especially for those transport or logistics companies with a specific customer portfolio. It can help planning the expansion related to the increase in the share of companies in the transport or logistics market. The interpretation of this indicator is discussed in the next section. ## 4 Discussion

Based on graph domination terms, we can proposed new parameters describing the transport networks and their bottlenecks/critical components:

1. Number of critical nodes = domination number;
2. Number of critical edges = edge-domination number;
3. Topological (structural) vulnerability of transport network = bondage-connected number;
4. Weighted structural vulnerability of transport network = weighted bondage-connected number;
5. Node-bottlenecks of transport network = dominating set, and weighted dominating set;
6. Arc-bottlenecks of transport network = edge-dominating set, and weighted edge-dominating set;
7. Consumer vulnerability = domination subdivision number. The first parameter gives information about how many nodes are essential in the transport network. Indirectly, we also get the smallest set of the most critical nodes in the system under consideration. This can help define the safety and security policy for this transport network. The second parameter should be treated similarly to the edges/arcs of the transport network. The next two parameters give information about how vulnerable the transport network is. These parameters inform how many (indirect also which) edges can be influenced by any threat before you need to increase the set of critical transport network nodes. The values of parameters determining the vulnerability of the transport network should be interpreted as follows. In the case of structural vulnerability and weighted structural vulnerability, the smaller amount of the number of bondage-connected and the weighted bondage-connected numbers, the more the transportation network is vulnerable, or, in other words, less resilient to a lack of passage through its nodes and arcs. The fifth and sixth parameters are closely related to the first and second parameters. They help in identifying the nodes and arcs crucial to the traffic engineering point of view. Furthermore, these two parameters form the basis for defined parameters of the transport network vulnerability. The bottlenecks can also be interpreted here as vulnerable to natural disasters or terrorist attacks. Thus, they are parameters supporting the policy of transport systems' safety and security at every scale. The last of the defined parameters is very useful in supply chain management or public transport planning. It indicates how much the considered system is susceptible to changes in the number of clients using given transport or logistics services. The customer vulnerability of the transport network is interpreted similarly to the topological vulnerability of the transport network. It means that the lower the value of the subdivision number is, the higher the customer vulnerability of this network is. As an example of using the methods proposed in section \"Results\" for determining weighted structural vulnerability, let us consider an intelligent transport system operating in a fixed part of the urban transport network, described by a vertex-weighted graph, presented in Figure 1. We assume that the weighted dominating set describes the most important crossroads along with an indication of the shortest time to cross it. In other words, the weight assigned to the vertices of the network determines the duration of the red light.