Reducible chains in several types of 2-connected graphs

Zhang, F. and X. Guo, Reducible chains in several types of 2-connected graphs, Discrete Mathematics 105 (1992) 285-291. Let F& 4, $ and 8 denote the sets of all 2-connected graphs, minimally 2-connected graphs, critically 2-connected graphs, and critically and minimally 2-connected graphs, respectively. We introduce the concept of %,-reducible chains of a graph G in %,, i = 0, 1, 2, 3, and give the upper bound and the lower bound of a number of ‘??z-reducible chains of G which are both sharp. Furthermore, a construction method of 4 is obtained. Let G = (V(G), E(G)) be a finite simple graph, and let K(G) be the connectivity of G. G is 2-connected if K(G) 3 2, G is minimally 2-connected if K(G) 2 2 but K(G e) < 2 for any e E E(G), and G is critically 2-connected if K(G) 3 2 but K(G V) < 2 for any u E V(G). We denote by 9j,, %i, Y& and Y& (= 9, tl$) the sets of all 2-connected graphs, minimally 2-connected graphs, critically 2-connected graphs, and critically and minimally 2-connected graphs, respectively. We call a vertex u critical if K(G) > 2 but K(G V) < 2. The cyclomatic number of G and the degree of a vertex v in G are denoted by v(G) and d,(v), respectively. A satisfactory construction method of F$ can be found in Tutte’s book [2]. Dirac gave a construction method of 3,. In this paper, by using the concept of Y&reducible chain, we obtain a method for constructing 3, i = 0, 1, 2, 3, and give the sharp upper and lower bounds of the number of Y&reducible chains. Definition 1. Let H be a subgraph of G. The graph induced by E(G) E(H) is denoted by G-H (i.e., E(G -H)=E(G)-E(H), and V(G-H)= {v) v is incident with an edge in E(G) E(H)}). * The project was supported by NSFC. 0012-365X/92/$05.00