The Split Systems Generated by the Bridges of a Connected Graph

The bridges of a connected graph G are shown to generate splits of G that are compatible in the sense that they generate a tree. The results are extended to gener- alized bridges, and a connection is made with cut-vertices and generalizations thereof. Further work is needed on the compatibility of splits determined by these generaliza- tions of bridges and cut-vertices. Hopefully the results will prove useful in the study of phylogenetic networks. 1 Notation The default assumption will be that any graph is nite and undirected. Let G be a graph, with V (G) its vertex set and E(G) its edge set. Denote the edge between x and y by the symbol xy. The vertices x and y are then the ends of the edge xy. We will assume a knowl- edge of the denitions of a path and a cycle. We shall use the notation (x1; x2; : : : ; xn; x1) to denote the cycle with edges x1x2; x2x3; : : : ; xn 1xn; xnx1. To say that G is connected is to say that there is a path between any pair of vertices. A connected component is a max- imal connected subgraph of G. Every graph is trivially the disjoint union of its connected components. Let k(G) denote the number of connected components of G, so k(G) = 1 for any connected graph G. We shall have occasion to remove selected edges or vertices from G. If xy is an edge, then G xy is the graph formed with V (G xy) = V (G) and E(G xy) = E(G) xy. If two edges are to be removed, we agree to let G xy; uv = (G xy) uv. For a vertex v, the graph G v is the graph formed by taking V (G v) = V (G) v, and removing any edge that involves v. We are now able to call xy a bridge if k(G xy) = k(G) + 1. This is of course a well known idea, and can be found in any standard text on graph theory. There are alternate equivalent denitions of a bridge. The most relevant one states that xy is a bridge if and only if it does not lie on any cycle. This is of course a well known idea, and can be found in any standard text on graph theory. Associated with the graph G, there is a graph L = L(G) called the line graph of G.