In many domains (e.g. Internet of Things, neuroimaging) signals are naturally supported on graphs. These graphs usually convey information on similarity between the values taken by the signal at the corresponding vertices. An interest of using graphs is that it allows to define ad hoc operators to perform signal processing. Among them, ones of paramount importance in many tasks are translations. In this paper we are interested in defining translations on graphs using a few simple properties. Namely we propose to define translations as functions from vertices to adjacent ones, that preserve neighborhood properties of the graph. We show that our definitions, contrary to other works on the subject, match usual translations on grid graphs.

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In this paper, we introduce translation operators on graphs. Contrary to spectrally-defined translations in the framework of graph signal processing, our operators mimic neighborhood-preserving properties of translation operators defined in Euclidean spaces directly in the vertex domain, and therefore do not deform a signal as it is translated. We show that in the case of grid graphs built on top of a metric space, these operators exactly match underlying Euclidean translations, suggesting that they completely leverage the underlying metric. More generally, these translations are defined on any graph, and can therefore be used to process signals on those graphs. We show that identifying proposed translations is in general an NP-Complete problem. To cope with this issue, we introduce relaxed versions of these operators, and illustrate translation of signals on random graphs.

In the 90's Clark, Colbourn and Johnson wrote a seminal paper, where they proved that maximum clique can be solved in polynomial time in unit disk graphs. Since then, the complexity of maximum clique in intersection graphs of (unit) d-dimensional balls has been investigated. For ball graphs, the problem is NP-hard, as shown by Bonamy et al. (FOCS '18). They also gave an efficient polynomial time approximation scheme (EPTAS) for disk graphs, however the complexity of maximum clique in this setting remains unknown. In this paper, we show the existence of a polynomial time algorithm for solving maximum clique in a geometric superclass of unit disk graphs. Moreover, we give partial results toward obtaining an EPTAS for intersection graphs of convex pseudo-disks.