This paper is concerned with terminable and interminable paths and trails
The only cohnected graphs which contain no 2 -way and in which no finite path is terminable are precisely all the 1m multiways.
Tho only connected graphs which have no 2 -m trail and in which no finite trail is terminable are precisely all the 1 -a multiways all of whose multiplicities are odd numbers and which have infinitely many bridges. In addition the structure of those connected graphs is determined which have a 1 -m trail and in which no 1 --m trail but every finite trail is terminable. In this paper the terminology and notation of a previous paper of the writer [l] and of I?. HARARY'S book [S] will be used. Furthermore, a graph consisting of the distinct nodes ni, . . . , n6 (where 8% 1) and of one or more (ni, n,+,)-edges for i = 1, . . . , 6 -1 will be called a multiway, and analogously for 1 -and 2m iiiultiways. The number of edges joining ni and n i f i will be called the (ni, nif1)multiplicity. Thus a inultiway in which each multiplicity is 1 is a way. Multiplicities are allowed to be infinite. in infinite graphs. It is shown that E incident with p . Then GY is finite, each node other than m has edge-degree 2n in Gr, and the edge-degree of rn in G: is ]El. Therefore IEI is even. This proves that G, has property (0). Since G, has (e), (g) and (h) it follows by Theorem 5 that G, has (a).
It may easily be verified that every graph belonging to family 2 has properties (e), (g) and (h), therefore by Theorem 5 it has (a).