A Graph with Central and Local Reference Set Yuni Listiana (a)(b), Liliek Susilowati (a*), Slamin (c)
(a) Mathematics Departement, Sains and Technology Faculty, Universitas Airlangga.
Jl. Dr. Ir. H. Soekarno Mulyorejo, Surabaya 60115, Indonesia
*liliek-s[at]fst.unair.ac.id
(b) Sains and Mathematics Education Departement, Universitas Dr Soetomo, Jalan Semolowaru No 84 Surabaya 60118, Indonesia
(c) Computer Sciences Departement, Computer science faculty, Universitas Jember
Jl. Kalimantan Tegalboto No.37, Jember 68121, Indonesia
Abstract
The local resolving set is a subset of vertex set in graph that introduced by Okamoto et al at 2010. It is refer to a subset where is for every two adjacent vertices in graph has different representation with the subset. Previously, in 1975 and 1988, Slater introduced the location set and reference set as another term of resolving set and minimum resolving set, respectively, and it was became a first time the idea of resolving set introduced. In this paper, we are develop a new concept refer to locating set and reference set by Slater and local resolving set by Okamoto, then combaining it with a central vertices of a simple and connected graph called as central local locating set and central local reference set. A new concept of a central local locating set and central local reference set are developed by observation and literature study of related concept. We used some class of graph, such as complete graph, complete bipartite graph, and sun graph to apply main definition of central local locating set and find their central local reference set. Then, the result will be formulate in a theorema that apply for general case. In addition, some lemmas is necessary need to support the proving of theorems in main results. The result are the central local reference set of complete graph and complete bipartite graph are equal with their vertex set. Meanwhile, the central local reference set of sun graph is equal with vertex set of its cycle. This is give a implication that since both of radius and diameter of complete graph and complete bipartite graph are equal so their central local reference set equal with their total vertex, meanwhile on sun graph both of radius and diameter different.
Keywords: Local locating set- local reference set- local metric dimension- central vertex- eccentricity
