禁止转移图和边着色图连通性问题的复杂性
The Complexity of Connectivity Problems in Forbidden-Transition Graphs and Edge-Colored Graphs
摘要：
禁止转移图的概念允许图中行走的鲁棒推广。在一个禁止变迁图中，每一对与一个公共顶点关联的边都是允许的或禁止的;如果一条行走上的所有连续边都是允许的，那么这条行走是相容的。禁止转移图及其相关模型在光通信网络、道路网络和生物信息学等领域有着广泛的应用。我们从参数化复杂性的角度开始研究基本的连通性问题，包括深入研究各种图宽度参数的易处理性。在几个结果中，我们证明了当用到线性森林的顶点删除距离参数化时，在禁止变迁图中找到给定端点之间的简单相容路径是W[1]-困难的（因此当用路径宽度或树宽度参数化时也是困难的）。另一方面，我们展示了一个代数技巧，当用树宽参数化找到一个适当着色的Hamilton圈时，它会产生易处理性;适当着色的边着色图中的行走是禁止过渡图中相容行走的最受研究的特殊情况之一。
英文摘要：
The notion of forbidden-transition graphs allows for a robust generalization of walks in graphs. In a forbidden-transition graph, every pair of edges incident to a common vertex is permitted or forbidden; a walk is compatible if all pairs of consecutive edges on the walk are permitted. Forbidden-transition graphs and related models have found applications in a variety of fields, such as routing in optical telecommunication networks, road networks, and bio-informatics. We initiate the study of fundamental connectivity problems from the point of view of parameterized complexity, including an in-depth study of tractability with regards to various graph-width parameters. Among several results, we prove that finding a simple compatible path between given endpoints in a forbidden-transition graph is W[1]-hard when parameterized by the vertex-deletion distance to a linear forest (so it is also hard when parameterized by pathwidth or treewidth). On the other hand, we show an algebraic trick that yields tractability when parameterized by treewidth of finding a properly colored Hamiltonian cycle in an edge-colored graph; properly colored walks in edge-colored graphs is one of the most studied special cases of compatible walks in forbidden-transition graphs.
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