# Graph Theory Approach to the Vulnerability of Transportation Networks
## 3 Results
### Structural Transportation Network Vulnerability
Then, the number of weighted bondage-connected numbergives information about the number of blocked network arcs that increase the number of crossroads requiring more attention and changing the duration of red light. The solution is considered in two cases:

1. Dynamic--after removing the arcs in graph, the new minimal dominating set has to be found;
2. Static--after removing the arcs in graph, the new minimal dominating set is the extension of the old one. In relation to the defined transportation network, the minimum weighted dominating set found by using Algorithm 4, is determined by the red vertices shown in Figure 2; e.g., \(D_{w}=\{B,D\}\). The weighted domination number is equal to 93. When solving a static problem, we assume that this set is the basis for a new minimal weighted dominating set. Using the weighted bondage-connected number as a parameter of weighted structural vulnerability, we can conclude that in case 1 (dynamic case) it is enough to remove the red dotted edge \(\{A,B\}\) for the weighted domination number to increase to 133 Figure 3.

However, if we consider a dynamic case in which after the edge removal operation, we are again looking for the minimal dominating set, then we can conclude that the value of the weighted structural vulnerability parameter is 1, and, as in the first case, removal of the red dotted edge \(\{A,B\}\) is enough to weighted domination number increased to 125 instead of the nominal 93, see as Figure 4.

Figure 1: Exemplary weighted transportation network; weights describe the red light time. Figure 3: Weighted bondage-connected number in transportation network with fixed weighted dominating set, \(wb_{c}(G)=1\). Figure 2: Minimal weighted dominating set in considered network, \(\gamma_{w}(G)=93\). ## 5 Conclusions

In the paper, six new vulnerability indicators of transport networks have been proposed. Every one of them is based on the domination concepts and related topics; e.g., bondage number and domination subdivision number. Two new algorithms have been introduced as the tools for calculating the values of the parameters defined. These algorithms will work well for small networks but may need more time for large-scale ones. The methods given in the article are universal and applicable to both small- and large-scale networks. Moreover, the presented approach can support the dynamic analysis of bottlenecks in transport networks. This research received funding from Gdynia Maritime University, project number WN\(\backslash\)Z\(\backslash\)03\(\backslash\)2019. The author declares no conflict of interest. ## References

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