A "walk" within a graph is usually defined for simple graphs, multigraphs or
even pseudographs as "alternating sequence of vertices and edges x0 , e1 ,
x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0 "walks on the edge level": For a positive integer s, an s-walk of length k
between hyperedges f and g is a sequence of hyperedges, f=e(0), e(1), ... ,
e(k)=g, where for j=1, ... , k, e(j-1) =/= e(j) and e(j-1) and e(j) have at
least s vertices in common (according to Aksoy et al.).
"walks on the vertex level": For a positive integer s, an s-walk of length k
between vertices a and b is a sequence of vertices, a=v(0), v(1), ... ,
v(k)=b, where for j=1, ... , k, v(j-1) and v(j) are connected by at least s
edges (analogous to Aksoy et al.).
There are two imperfections for the definition for "walks on the edge level":
one is that a walk of length 1 consists of two edges (or a walk of length 0
consists of one edge), whereas a walk of length 1 on the vertex level
consists of two vertices and one edge (or a walk of length 0 consists of one
vertex and no edge). The other is that edges, especially loops, can be
traversed only once (and not repeatedly) because of the condition
e(j-1) =/= e(j). The latter is avoided in the definition for ,
see df-ewlks. To be compatible with the (usual) definition of walks for
pseudographs, walks also suitable for arbitrary hypergraphs are defined "on
the vertex level" in the following as , see df-wlks,
restricting s to 1. wlk1ewlk shows that such a 1-walk "on the vertex
level" induces a 1-walk "on the edge level".