Title: Critical Vertices and Edges in $H$-free Graphs,
Abstract: 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.