Abstract: This paper proved the following results:(1) The ⊙-graph of a 2-connected graph is 2(p-1) connected; (2) if a 2-connected graph G has two unicyclic spanning subgraphs and the number of one-valent vertices of these two subgraphs are respectively m and n (m<n), then for any integer k:m≦k≦n,there exist at least 2(p-1) unicyclic subgraphs of graph G whick have k one-valent vertices. Here, p is the dimension of the cycle space of G.

Key words: 2-connected graph, unicyclic spanning subgraph, ⊙-graph, adjacent ⊙-graph, cut edge, 2-connected graph, unicyclic spanning subgraph, ⊙-graph, adjacent ⊙-graph