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A mobile agent, modeled as a deterministic finite automaton, navigates in theinfinite anonymous oriented grid $\mathbb{Z} \times \mathbb{Z}$. It has toexplore a given infinite subgraph of the grid by visiting all of its nodes. Wefocus on the simplest subgraphs, called {\em wedges}, spanned by all nodes ofthe grid located between two half-lines in the plane, with a common origin.Many wedges turn out to be impossible to explore by an automaton that cannotmark nodes of the grid. Hence, we study the following question: Given a wedge$W$, what is the smallest number $p$ of (movable) pebbles for which thereexists an automaton that can explore $W$ using $p$ pebbles? Our maincontribution is a complete solution of this problem. For each wedge $W$ wedetermine this minimum number $p$, show an automaton that explores it using $p$pebbles and show that fewer pebbles are not enough. We show that this smallestnumber of pebbles can vary from 0 to 3, depending on the angle betweenhalf-lines limiting the wedge and depending on whether the automaton can crossthese half-lines or not.

Exploring Wedges of an Oriented Grid by an Automaton with Pebbles