Paper Special Section on Discrte Mathematics and Its Applications Is a given Flow Uncontrollable?

SUMMARY An s-t ow in a directed network is called \un-controllable", when the ow is representable as a positive sum of elementary s-t path ows. In this paper, we discuss the problem \Is a given ow uncontrollable?". We show that the problem is NP-complete. 1. Introduction Recently, Iri proposed a new type of network ow theory [1], where only those ows which are representable as a positive sum of elementary path ows are considered. The ows in this framework are called uncontrollable ows. In uncontrollable network ow theory, we do not assume the controllability of ows, but will admit uncontrolled and/or emergency ows to take place in a network. In this paper, we discuss the problem for determining whether a given ow is uncontrollable or not. This problem was presented in the paper [1]. If a trac ow is an uncontrollable ow, it seems that the ow will avoid the congestion or undesirable phenomena even in the case of emergency. The problem aords a foundation for designing a network which prevents the congestion when the users are not obedient. Now, we give a brief explanation of uncontrollable ows. A two-terminal network is a network ((V; E); s; t) with a vertex set V; a directed edge set E and two vertices s and t specied as the source and sink, respectively. An elementary s-t path is a directed path from s to t which traverses each vertex at most once. An s-t ow in a two-terminal