无$H$图的临界点和临界边
Critical Vertices and Edges in $H$-free Graphs
摘要：
如果图中的顶点或边的删除使图的色数减少1，则该顶点或边是临界的。我们考虑的问题，决定是否有一个图的临界点或边，分别。对于这两个问题，我们给出了一个复杂性二分法，它仅限于$H$ -free图，即没有导出子图同构于$H$的图.此外，我们证明了边是临界的，当且仅当它的收缩使色数减少1。因此，我们也得到了一个复杂性二分法的问题，决定如果一个图有一个边的收缩减少色数1。
英文摘要：
A vertex or edge in a graph is critical if its deletion reduces the chromatic number of the graph by 1. We consider the problems of deciding whether a graph has a critical vertex or edge, respectively. We give a complexity dichotomy for both problems restricted to $H$-free graphs, that is, graphs with no induced subgraph isomorphic to $H$. Moreover, we show that an edge is critical if and only if its contraction reduces the chromatic number by 1. Hence, we also obtain a complexity dichotomy for the problem of deciding if a graph has an edge whose contraction reduces the chromatic number by 1.
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