Abstract
The geodetic number of a graph is an important graph invariant. In 2002, Atici showed the geodetic set determination of a graph is an NP-Complete problem. In this paper, we compute the geodetic set and geodetic number of an important class of graphs called the k-th power of a cycle. This class of graphs has various applications in Computer Networks design and Distributed computing. The k-th power of a cycle is the graph that has the same set of vertices as the cycle and two different vertices in the k-th power of this cycle are adjacent if the distance between them is at most k.
Highlights
IntroductionAll graphs are simple (finite, loopless, undirected, and without multiple edges), the vertex set and edge set of such graphs are denoted by V ( G ) and E( G ), respectively
In this paper, all graphs are simple, the vertex set and edge set of such graphs are denoted by V ( G ) and E( G ), respectively
For a vertex u of the graph G, we define the eccentricity of u in G (denoted by e(u)) to be the maximum of the distances between u and all other vertices of G [1]
Summary
All graphs are simple (finite, loopless, undirected, and without multiple edges), the vertex set and edge set of such graphs are denoted by V ( G ) and E( G ), respectively. For two different vertices u and v of a graph G, the distance between u and v in G (denoted by dG (u, v)) is defined to be the length of the shortest path between u and v. G is defined by the graph that has the same set of vertices as G and two different vertices in the k-th power of G are adjacent if the distance between them is at most k. For any simple graph G = (V ( G ), E( G )) and positive integer k, the k-th power of G (denoted by G k ) is the simple graph with vertex set V ( G k ) = V ( G ) and edge set E( G k ) = {uv : u, v ∈ E( G ) and 0 < dG (u, v) ≤ k }, obviously G1 = G.
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