Topological complexity of configuration spaces of fully articulated graphs and banana graphs (Q2664107)

The ordered configuration space of \(n\) particles on some graph \(G\) is denoted by \(\mathrm{Conf}_{n}(G)\) and the (normalized) topological complexity of a space \(X\) is denoted by \(\mathrm{TC}(X)\). Let \(G\) be a graph with \(\beta\) bifurcations, i.e. vertexes with valence at least 3. Farber has shown that if \(G\) is a tree and \(n \ge 2\beta\), then \(\mathrm{TCConf}_{n}(G) = 2\beta\). A graph is {\em fully articulated} if removing any bifurcation makes it disconnected. The authors extend Farber's result to fully articuled graphs. The authors also complete Farber's result showing that if \(G\) is a tree which is not homeomorphic to the letter Y, then \(\mathrm{TCConf}_{n}(G) = 2 \min\{\lfloor n/2 \rfloor, \beta\}\). The {\em banana graph} on \(k\ge 1\) edges, denoted by \(B_k\), is the graph consisting of two vertices connected by \(k\) edges. The authors show that \(\mathrm{TCConf}_{n}(B_k) = 4\) if \(k \ge 4\) and \(n \ge 3\), and that \(\mathrm{TCConf}_n(B_k) = 2\) if \(k \ge 3\) and \(n \le 2\), or \(k = n = 3\). Actually Farber conjectured that the equality \(\mathrm{TCConf}_n(G) = 2\beta\) holds for any connected graph \(G\) with \(\beta \ge 2\) and \(n \ge 2\beta\). In the last part of the article, the authors show that this conjecture does not hold for unordered configuration spaces instead of ordered ones.