Given a graph H, a graph G is called H-critical if G does not admit a homomorphism to H, but any proper subgraph of G does. Observe that K k−1-critical graphs are the standard k-(colour)-critical graphs. We consider questions of extremal nature previously studied for k-critical graphs and generalize them to H-critical graphs. After complete graphs, the next natural case to consider for H is that of the odd-cycles. Thus, given integers and k, ≥ k, we ask: what is the smallest order of a C 2 +1-critical graph of odd-girth at least 2k + 1? Denoting this value by η(k, C 2 +1), we show that η(k, C 2 +1) = 4k for 1 ≤ ≤ k ≤ 3 +i−3 2 (2k = i mod 3) and that η(3, C 5) = 15. The latter means that a smallest graph of odd-girth 7 not admitting a homomorphism to the 5-cycle is of order 15. Computational work shows that there are exactly eleven such graphs on 15 vertices.