Partial confluence and closed subsets of a graph (Q690291)

The author works within a general setting of continua theory and proves a very interesting result for graphs. Even more, he mentions an interesting application of his main result in computer networks, which are graphs. In this work weakly confluent maps as well as \(w\)-sets play an essential role. If \(f : X \to Y\) is an onto-mapping between continua, then a subcontinuum \(K\) of \(Y\) is a \(w_ f\)-set provided that there is a subcontinuum \(K'\) of \(X\) mapped onto \(K\). If \(K\) is a closed subset of \(Y\), then \(P(K)\) is the smallest cardinal such that for each map \(f\) of a continuum \(X\) onto \(Y\), \(K\) is the union of at most \(P(K)\) \(w_ f\)-sets. If \(G\) is a graph and \(K\) a closed subset of \(G\) with dense interior, then \(P(K) = 2 [\beta (G) - \beta (\text{cl} (G \backslash K))] + c [\text{cl} (G \backslash K)] + t(K)-1\), where \(\beta (G)\) is the first Betti number of \(G\), \(c [\text{cl} (G \backslash K)]\) is the number of components of \(\text{cl} (G \backslash K)\) and \(t(K)\) the number of terminal points of \(G\) in \(K\). In the last section of the paper the author proves an extension of a theorem by Grispolakis-Nadler and Tymchatyn and he shows that if \(f : M \to Y\) is a monotone mapping between continua, \(K\) is a subcontinuum of \(Y\) and \(f\) is one-to-one on a dense subset of \(f^{-1} (K)\), then \(P(f^{-1} (K)) = P(K)\). Finally, he characterizes, among graphs, those for which \(P(K) = n\) for every nondegenerate subcontinuum \(K\). We should mention that the reviewer and \textit{E. D. Tymchatyn} [Houston J. Math. 5, 483-502 (1979; Zbl 0412.54039); 6, 375-387 (1980; Zbl 0447.54042)] have characterized completely the class of all those continua for which \(P(K) = 1\).