Three Tree-Paths zy Avram Zehavi Alon ltai z COMPUTER SCIENCE DEPARTMENT ECHNION- /IT, HAIFA, ISRAEL ABSTRACT ltai and Rodeh [3] have proved that for any 2-connected graph G and any vertex s E G there are two spanning trees such that the paths from any other vertex to s on the trees are disjoint. In this paper the result is gen- eralized to 3-connected graphs. 1. INTRODUCTION zyxwvu A zyxwvutsrq graph G = (V(G),E(G)) zyxwv consists of a set of vertices V(G) and a set E(G) of unordered pairs of V(G) called edges. An x - y path .rr[x,y] in a graph G is a sequence of distinct vertices of V(G) (x = zyxw uo, zyxw ul, . . . , u, = y) such that (u,-,, u,) E E(G) (i = 1, . . . , zyx m). Two paths .rr,[x,,y,] and .rr,[x,,y,] are dis- joint if they have no common vertices except perhaps {x,,yl} fl {x2,y2}. Let K(X, y; G) denote the maximum number of disjoint x - y paths in G. A graph G is k-connected if between any two vertices of G there are at least k disjoint paths. The connectivity K(G) of G is the maximum integer k such that G is k-connected, i.e., min{K(x,y;G): x,y E V(G)} = k. If u E V(G), we denote by T(u) the set of all vertices adjacent to u while T,(u) denotes the set of edges adjacent to u. The degree of u, deg(u;G), is equal to the cardinality of rE(u). A vertex u is a leaf if deg(u; G) = 1. We abuse the notation and say that an edge or a vertex belongs to a graph (e E G or u E G instead of e E E(G) or u E V(G)). Also, G - e denotes the graph from which e was deleted from the edge set (likewise G + e). For all other notations refer to Bondy and Murty [ 11. Let T be a spanning tree of G and u, w E V(G): then T[u, w] is the path from u to w on T. The following result appears in [3]: Theorem 1.1. If G is a 2-connected graph and zyx s E V(G), then there exist two spanning trees of G, TI, T, such that for every vertex u E G the paths T,[u, s] and T,[u, s] are disjoint. Itai and Rodeh [3] used this result to develop distributed algorithms that are resilient to the failure of a single line (or processor). To increase reliability, more spanning trees are needed. The purpose of this paper is to prove the following: Journal of Graph Theory, Vol. 13, No. 2, 175-188 (1989) 0 1989 by John Wiley & Sons, Inc. CCC 0364-902418910201 75-14$04.00