Generalized uniformly line vulnerable digraphs
For two vertices u and v of a graph G, the closed interval I[u,v] consists of u, v, and all vertices lying in some u-v geodesic in G. If S is a set of vertices of G, then I[S] is the union of all sets I[u,v] for u, v ∈ S. If I[S] = V(G), then S is a geodetic set for G. The geodetic number g(G) is the minimum cardinality of a geodetic set. A set S of vertices in a graph G is uniform if the distance between every two distinct vertices of S is the same fixed number. A geodetic set is essential if for...
We study the trajectory of a simple random walk on a d-regular graph with d ≥ 3 and locally tree-like structure as the number n of vertices grows. Examples of such graphs include random d-regular graphs and large girth expanders. For these graphs, we investigate percolative properties of the set of vertices not visited by the walk until time un, where u > 0 is a fixed positive parameter. We show that this so-called vacant set exhibits a phase transition in u in the following sense: there...