This paper deals with the problem of enumerating 3-edge-connected spanning subgraphs of an input plane graph. In 2018, Yamanaka et al. proposed two enumeration algorithms for such a problem. Their algorithm generates each 2-edge-connected spanning subgraph of a given plane graph with n vertices in O(n) time, and another one generates each k-edge-connected spanning subgraph of a general graph with m edges in O(mT) time, where T is the running time to check the k-edge connectivity of a graph. This paper focuses on the case of the 3-edge-connectivity in a plane graph. We give an algorithm which generates each 3-edge-connected spanning subgraph of the input plane graph in O(n2) time. This time complexity is the same as the algorithm by Yamanaka et al., but our algorithm is simpler than theirs.

In this paper, we consider the problem of enumerating spanning subgraphs with high edge-connectivity of an input graph. Such subgraphs ensure multiple routes between two vertices. We first present an algorithm that enumerates all the 2-edge-connected spanning subgraphs of a given plane graph with n vertices. The algorithm generates each 2-edge-connected spanning subgraph of the input graph in O(n) time. We next present an algorithm that enumerates all the k-edge-connected spanning subgraphs of a given general graph with m edges. The algorithm generates each k-edge-connected spanning subgraph of the input graph in O(mT) time, where T is the running time to check the k-edge-connectivity of a graph.

Let Ni be the number of connected spanning subgraphs with i(n-1 i m) edges in an n-vertex m-edge undirected graph G=(V,E). Although Nn-1 is computed in polynomial time by the Matrix-tree theorem, whether Nn is efficiently computed for a graph G is an open problem (see e.g., [2]). On the other hand, whether Nn2≥ Nn-1Nn+1 for a graph G is also open as a part of log concave conjecture (see e.g., [6],[12]). In this paper, for a complete graph Kn, we give the formulas for Nn, Nn+1, by which Nn, Nn+1 are respectively computed in polynomial time on n, and, in particular, prove Nn2> Nn-1Nn+1 as well.

Consider an undirected multigraph G=(V,E) with n vertices and m edges, and let Ni denote the number of connected spanning subgraphs with i(min) edges in G. Recently, we showed in [3] the validity of (m-i+1)Ni-1>Ni for a simple graph and each i(min). Note that, from this inequality, 2 is easily derived. In this paper, for a multigraph G and all i(min), we prove (m-i+1)Ni-1(i-n+2)Ni, and give a necessary and sufficient condition by which (m-i+1)Ni-1=(i-n+2)Ni. In particular, this means that (m-i+1)Ni-1>Ni is not valid for all multigraphs, in general. Furthermore, we prove 2, which is not straightforwardly derived from (m-i+1)Ni-1(i-n+2)Ni, and also introduce a necessary and sufficent condition by which =2. Moreover, we show a sufficient condition for a multigraph to have Nn2>Nn-1Nn+1. As special cases of the sufficient condition, we show that if G contains at least +1 multiple edges between some pair of vertices, or if its underlying simple graph has no cycle with length more than 4, then Nn2>Nn-1Nn+1.

Consider an undirected graph G=(V,E) with n (=|V|) vertices and m (=|E|) edges. It is well-known that the problem of computing the sequence Nn-1,Nn,...,Nm is #P-complete (see e.g.,[3]), where Ni denotes the number of connected spanning subgraphs with i (n-1!im) edges in G. In this paper, by proving new inequalities on the sequence Nn-1,Nn,...,Nm, we show an interesting and stronger property that the sequence γn-1,γn,...,γm, where γi denotes the average number of spanning trees in the connected spanning subgraphs with i edges, is a convex sequence as well as a monotonically increasing sequence, although this property does not hold for the sequence Nn-1,Nn,...,Nm.

We consider the problem of finding a minimum weight k-connected spanning subgraph of a given edge-weighted graph G for k=3. The problem is known to be NP-hard for k 2, and there are an O(n2m) time 3-approximation algorithm due to Nutov and Penn and an O(n8) time 2-approximation algorithm due to Dinitz and Nutov, where n and m are the numbers of vertices and edges in G, respectively. In this paper, we present a 7/3-approximation algorithm which runs in O(n2m) time.