A polynomial algorithm for cyclic edge connectivity of cubic graphs

In this paper, we develop a polynomial time algorithm to find out all the minilnum cyclic edge cutsets of a 3-regular graph, and therefore to determine the cyclic edge connectivity of a cubic graph. The algorithm is recursive, with complexity bounded by O(n31og2 n). The algorithm shows that the number of mini~um cyclic edge cut sets of a 3-regular graph G is polynornial in v( G) and that the minimum cyclic edge cutsets can be found in polynomial time, and so the cyclic edge connectivity of G can be calculated. O. Introduction For a connected graph G, a vertex set S is said to be a vertex cutset of G, if G-S is not connected. The connectivity K(G) of G is the minimim cardinality of all the vertex cutsets of G Similarly, an edge cutset E of G is an edge set such that G-E is not connected, and the edge connectivity A.(G) of G is the minimum cardinality of all the edge cutsets of G Here we are going to discuss another type of connectivity of a graph, the cyclic edge connectivity, which is defined below. For a connected graph G, a cyclic edge cutset is an edge cutset whose deletion disconnects the graph and such that two of the components created each must contain at least one cycle. The cyclic edge connectivity CA(G) is the minimum cardinality of all the cyclic edge cutsets of G If no cyclic edge cutset exists, we set clt(G) =0. In this paper, we consider only simple, undirected graphs. All terminology and notation not defined in the paper can be found in [2]. The concept of cyclic edge connectivity was introduced by Tait[ 1 0] and studied, in particular, by Plummer[8], for planar graphs, with a slight difference: if no cyclic edge cutset exists, they set CA.-(G)==. In references [4), [5] and [7], the relation between cyclic edge connectivity and n-extendability of graphs is studied. In a paper of Peroche[9], several sorts of connectivity, including cyclic edge connectivity, and their relations are studied. The following upper bound for the cyclic edge connectivity of a graph G is given there. Australasian Journal of Combinatorics 24(2001). pp.247-259 Theorem 1: If G=(V, E) is a simple graph with IVI=n, then CA-(C) ~ 3(n-3), for n;::: 6, and the bound is sharp. Equality holds when G=Kn. How to compute the cyclic edge connectivity of an arbitrary graph has not been studied in the literature as far as we know. Even for cubic graphs, the distribution of minimum cyclic edge cutsets is unknown. If there are polynomially many such cutsets in a cubic graph, is it possible to find them in polynomial time? This paper presents a recursive algorithm to find all the cyclic edge cutsets of an input cubic graph, with time complexity bounded by O(n3log2n). In this paper, we are going to develop a polynomial time algorithm that computes the cyclic edge connectivity of a cubic graph. We use the concept of removing an edge from a 3-connected graphs. This was introduced and studied by Barnette and Grtinbaum[ 1]. The distribution of removable edges in 3-connected graphs was studied by Holton, Jackson, Saito and Wormald[3]. In the first and second sections, we introduce a necessary and sufficient condition for a cubic graph to have a cyclic edge cutset. Then the concept of removing an edge from a 3-connected graph is presented, and how the removed edge is used in the algorithm to help compute the cyclic edge connectivity is discussed. In the third section, an algorithm that returns all the minimum cyclic edge cutsets of a given cubic graph is described. In the fourth section, we give an example of applying the algorithm. We find all the minimum cyclic edge cutsets of the Petersen graph, and show that the cyclic edge connectivity of the Petersen graph is 5. In the last section, the time complexity of the recursive program is analysed. 1. Preliminaries Firstly, we give a necessary and sufficient condition for a cubic graph to have a cyclic edge cutset. Theorem 2: Let G be a 3-regular graph of order u, let g be the girth of G. Then G has a cyclic edge cutset if and only if v > 2g 2 . Proof. If G has a cyclic edge cutset S, then let C be one of the cycles in G-S of length c. By the definition of cyclic edge cutset, G-V(C) must contain at least one cycle. Then, we have 3( v c) c > 2( v c 1) v > 2c 2 ~ 2g 2 . Conversely, if v> 2g 2, let C be a minimum cycle in G, then 3(v-g)-g > 2(v-g-l). hence G-V(C) must contain at least one cycle and (V(C), V(G)\V(C» is a cyclic edge