一些圈的顶点合并的划分维度
The partition dimension of the vertex amalgamation of some cycles.
作者信息
Hinding Nurdin, Nurwahyu Budi, Syukur Daming Ahmad, Kamal Amir Amir
机构信息
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Hasanuddin University, Indonesia.
出版信息
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)。
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