On a problem concerning ordered colourings

Let G be a connected graph with v(G) ⩾ 2 vertices and independence number α(G). G is critical if for any edge e of G: (i) α(G − e) >α(G), if e is not a cut edge of G, and (ii) v(Gi) − α(Gi) < v(G)− α(G), i = 1, 2, if e is a cut edge and G1, G2 are the two components of G − e. Recently, Katchalski et al. (1995) conjectured that: if G is a connected critical graph, then image with equality possible if and only if G is a tree. In this paper we establish this conjecture.