Stand-up indulgent gathering on lines for myopic luminous robots

Abstract We consider a strong variant of the crash fault-tolerant gathering problem called Stand-Up Indulgent Gathering (SUIG), by robots endowed with limited visibility sensors and lights on line-shaped networks. In this problem, a group of mobile robots must eventually gather at a single location, not known beforehand, regardless of the occurrence of crashes. Differently from previous work that considered unlimited visibility, we assume that robots can observe nodes only within a certain fixed distance (i.e. they are myopic), and emit a visible color from a fixed set (i.e. they are luminous), without multiplicity detection. We consider algorithms depending on two parameters related to the initial configuration: $M_{\mathrm{init}}$, which denotes the number of nodes between two border nodes, and $O_{\mathrm{init}}$, which denotes the number of nodes hosting robots. Then, a border node is a node hosting one or more robots that cannot see other robots on at least one side. Our main contribution is to prove that, if $M_{\mathrm{init}}$ or $O_{\mathrm{init}}$ is odd, SUIG can be solved in the fully synchronous model even if crashes occur at a node at any phase of execution (i.e. including within the LCM cycle).