# The minimum vulnerability problem on specific graph classes

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## Abstract

Suppose that each edge *e* of an undirected graph *G* is associated with three nonnegative integers \(\mathsf{cost}(e)\), \(\mathsf{vul}(e)\) and \(\mathsf{cap}(e)\), called the cost, vulnerability and capacity of *e*, respectively. Then, we consider the problem of finding \(k\) paths in *G* between two prescribed vertices with the minimum total cost; each edge *e* can be shared without any cost by at most \(\mathsf{vul}(e)\) paths, and can be shared by more than \(\mathsf{vul}(e)\) paths if we pay \(\mathsf{cost}(e)\), but cannot be shared by more than \(\mathsf{cap}(e)\) paths even if we pay the cost for *e*. This problem generalizes the disjoint path problem, the minimum shared edges problem and the minimum edge cost flow problem for undirected graphs, and it is known to be NP-hard. In this paper, we study the problem from the viewpoint of specific graph classes, and give three results. We first show that the problem is NP-hard even for bipartite outerplanar graphs, 2-trees, graphs with pathwidth two, complete bipartite graphs, and complete graphs. We then give a pseudo-polynomial-time algorithm for bounded treewidth graphs. Finally, we give a fixed-parameter algorithm for chordal graphs when parameterized by the number \(k\) of required paths.

## Keywords

Bounded treewidth graph Chordal graph Fixed parameter tractability Graph algorithm Minimum vulnerability problem## Notes

### Acknowledgments

Magnús M. Halldórsson and Christian Konrad are supported by Icelandic Research Fund Grant-of-Excellence No. 120032011. Takehiro Ito is partially supported by JSPS KAKENHI 25330003.

### Compliance with ethical standards

### Conflict of Interest

The authors declare that they have no conflict of interest.

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