  Moving an autonomous agent through an unknown environment is one of the
crucial problems for robotics and network analysis. Therefore, it received a
lot of attention in the last decades and was analyzed in many different
settings. The graph exploration problem is a theoretical and abstract model,
where an algorithm has to decide how the agent, also called explorer, moves
through a network such that every point of interest is visited at least once.
For its decisions, the knowledge of the algorithm is limited by the perception
of the explorer.
  There are different models regarding the perception of the explorer. We look
at the fixed graph scenario proposed by Kalyanasundaram and Pruhs (Proc. of
ICALP, 1993), where the explorer starts at a vertex of the network and sees all
reachable vertices, their unique names and their distance from the current
position. Therefore, the algorithm recognizes already seen vertices and can
adapt its strategy during exploring, because it does not forget anything.
  Because the algorithm only learns the structure of the graph during
computation, it cannot deterministically compute an optimal tour that visits
every vertex at least once without prior knowledge. Therefore, we are
interested in the amount of crucial a-priori information needed to solve the
problem optimally, which we measure in terms of the well-studied model of
advice complexity. [..]
  We look at different variations of the graph exploration problem and
distinguish between directed or undirected edges, cyclic or non-cyclic
solutions, unit costs or individual costs for the edges and different amounts
of a-priori structural knowledge of the explorer. [..] In this work, we present
algorithms with an advice complexity of $\mathcal{O}(m+n)$, thus improving the
classical bound for sparse graphs.
