We consider a bistable reaction-diffusion equation on a metric graph that is a generalization of the so-called star graphs. More precisely, our graph \(\Omega\) consists of a bounded finite metric graph D of arbitrary configuration and a finite number of branches \(\Omega _1,\ldots ,\Omega _N\,(N\ge 2)\) of infinite length emanating from some of the vertices of D. Each \(\Omega _i\,(i=1,\ldots ,N)\) is called an “outer path”. Our goal is to investigate the behavior of the front coming from infinity along a given outer path \(\Omega _i\) and to discuss whether or not the front propagates into other outer paths \(\Omega _j\,(j\ne i)\) . Unlike the case of star graphs, where D is a single vertex, the dynamics of solutions can be far more complex and may depend sensitively on the configuration of the center graph D. We first focus on general principles that hold regardless of the structure of the center graph D. Among other things, we introduce the notion “limit profile”, which allows us to define “propagation” and “blocking” without ambiguity, then we prove transient properties, that is, propagation \(\Omega _i\rightarrow \Omega _j\) and \(\Omega _j\rightarrow \Omega _k\) imply propagation \(\Omega _i\rightarrow \Omega _k\) . Next we consider perturbations of the graph D while fixing the outer paths \(\Omega _1,\ldots ,\Omega _N\) and prove that if, for a given choice of i, j, propagation \(\Omega _i\rightarrow \Omega _j\) occurs for a graph D, then the same holds for any graph \(D'\) that is sufficiently close to D (robustness under perturbation). We also consider several specific classes of graphs, such as those with a “reservoir” type subgraph, and study their intriguing properties.
