Ying WANG, Xin Min HOU
Let γc(G) denote the connected domination number of a graph G. A graph G is called k-connected-domination-critical, or briefly k-c-critical, if γc(G) = k, but γc(G+e) ≤ k - 1 for any edge e ∈ E(G). A graph G is factor-bicritical, if G-u-v has a perfect matching for every pair of distinct vertices u, v ∈ V (G). In this paper, following the work of Ananchuen, Ananchuen and Plummer [Matching properties in connected domination critical graphs, Discrete Math., 2008], we show that: (1) If a graph G is 3-connected 3-c-critical of order 2n (n ≥ 4) satisfying d(x) + d(y) ≥ 2n - 2 for any two nonadjacent vertices x, y in G, then G is factor-bicritical, this generalizes a result of Ananchuen, Ananchuen and Plummer (Theorem 2.1 in their paper). (2) Let G be a 3-connected 3-c-critical graph of order 2n (n ≥ 4). If for any two distinct vertices u and v in G with distance two, we have max{d(u), d(v)} ≥ n, then G is factor-bicritical. (3) If G is a 3-connected 3-c-critical K1,4-free graph of even order, then G is factorbicritical, this improves a result of Ananchuen, Ananchuen and Plummer (Theorem 2.2 in their paper).