Hinding Nurdin, Nurwahyu Budi, Syukur Daming Ahmad, Kamal Amir Amir
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Hasanuddin University, Indonesia.
Heliyon. 2022 May 31;8(6):e09596. doi: 10.1016/j.heliyon.2022.e09596. eCollection 2022 Jun.
Let be a connected, finite, simple, and undirected graph. The distance between two vertices , denoted by , is the shortest length of - in . The distance between a vertex is defined as where , denoted by . For an ordered partition of the vertices of a graph , the partition representation of a vertex with respect to Π is defined as the - . The partition set Π is called a resolving partition of , if , for all , . The partition dimension of is the minimum number of sets in any resolving partition of . In this paper we study the partition dimension of the vertex amalgamation of some cycles. Specifically, we present the vertex amalgamation of copies of the cycle at a fixed vertex , for and , .
设(G)是一个连通、有限、简单且无向的图。两个顶点(u)和(v)之间的距离，记为(d(u, v))，是(G)中从(u)到(v)的最短路径长度。顶点(v)到顶点集(S)的距离定义为(\min{d(v, u) : u \in S})，记为(d(v, S))。对于图(G)顶点的有序划分(\Pi = {V_1, V_2, \ldots, V_k})，顶点(v)关于(\Pi)的划分表示定义为((d(v, V_1), d(v, V_2), \ldots, d(v, V_k)))。如果对于所有(u, v \in V(G))，(u \neq v)，都有((d(u, V_1), d(u, V_2), \ldots, d(u, V_k)) \neq (d(v, V_1), d(v, V_2), \ldots, d(v, V_k)))，则划分集(\Pi)称为(G)的分辨划分。(G)的划分维数是任何分辨划分中集合的最小数量。在本文中，我们研究一些圈的顶点合并的划分维数。具体来说，我们给出了在固定顶点(v)处(m)个圈(C_n)的顶点合并，其中(m \geq 2)且(n \geq 3)。
