The partition dimension of the vertex amalgamation of some cycles
Let G=(V(G),E(G)) be a connected, finite, simple, and undirected graph. The distance between two vertices u,w∈V(G), denoted by d(u,w), is the shortest length of (u,w)-path in G. The distance between a vertex v∈V(G) is defined as min{d(v,x):x∈S} where S⊂V(G), denoted by d(v,S). For an ordered partit...
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| Format: | Article |
| Language: | English |
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Elsevier
2022-06-01
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| Series: | Heliyon |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2405844022008842 |
| _version_ | 1811334990952136704 |
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| author | Hasmawati Nurdin Hinding Budi Nurwahyu Ahmad Syukur Daming Amir Kamal Amir |
| author_facet | Hasmawati Nurdin Hinding Budi Nurwahyu Ahmad Syukur Daming Amir Kamal Amir |
| author_sort | Hasmawati |
| collection | DOAJ |
| description | Let G=(V(G),E(G)) be a connected, finite, simple, and undirected graph. The distance between two vertices u,w∈V(G), denoted by d(u,w), is the shortest length of (u,w)-path in G. The distance between a vertex v∈V(G) is defined as min{d(v,x):x∈S} where S⊂V(G), denoted by d(v,S). For an ordered partition Π={S1,S2,…,Sk} of the vertices of a graph G, the partition representation of a vertex v∈V(G) with respect to Π is defined as the k-vektor r(v|Π)=(d(v,S1),d(v,S1),…,d(v,S1). The partition set Π is called a resolving partition of G, if r(u|Π)≠r(v|Π), for all u≠v, u,v∈V(G). The partition dimension of G is the minimum number of sets in any resolving partition of G. In this paper we study the partition dimension of the vertex amalgamation of some cycles. Specifically, we present the vertex amalgamation of m copies of the cycle Cn at a fixed vertex v∈V(Cn), for n≥6 and k2−3k+42≤m≤k2−k2, k≥3. |
| first_indexed | 2024-04-13T17:17:21Z |
| format | Article |
| id | doaj.art-2b16337d2b544b408d41acd46873cf8a |
| institution | Directory Open Access Journal |
| issn | 2405-8440 |
| language | English |
| last_indexed | 2024-04-13T17:17:21Z |
| publishDate | 2022-06-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Heliyon |
| spelling | doaj.art-2b16337d2b544b408d41acd46873cf8a2022-12-22T02:38:06ZengElsevierHeliyon2405-84402022-06-0186e09596The partition dimension of the vertex amalgamation of some cycles Hasmawati0Nurdin Hinding1Budi Nurwahyu2Ahmad Syukur Daming3Amir Kamal Amir4Department of Mathematics, Faculty of Mathematics and Natural Sciences, Hasanuddin University, IndonesiaCorresponding author.; Department of Mathematics, Faculty of Mathematics and Natural Sciences, Hasanuddin University, IndonesiaDepartment of Mathematics, Faculty of Mathematics and Natural Sciences, Hasanuddin University, IndonesiaDepartment of Mathematics, Faculty of Mathematics and Natural Sciences, Hasanuddin University, IndonesiaDepartment of Mathematics, Faculty of Mathematics and Natural Sciences, Hasanuddin University, IndonesiaLet G=(V(G),E(G)) be a connected, finite, simple, and undirected graph. The distance between two vertices u,w∈V(G), denoted by d(u,w), is the shortest length of (u,w)-path in G. The distance between a vertex v∈V(G) is defined as min{d(v,x):x∈S} where S⊂V(G), denoted by d(v,S). For an ordered partition Π={S1,S2,…,Sk} of the vertices of a graph G, the partition representation of a vertex v∈V(G) with respect to Π is defined as the k-vektor r(v|Π)=(d(v,S1),d(v,S1),…,d(v,S1). The partition set Π is called a resolving partition of G, if r(u|Π)≠r(v|Π), for all u≠v, u,v∈V(G). The partition dimension of G is the minimum number of sets in any resolving partition of G. In this paper we study the partition dimension of the vertex amalgamation of some cycles. Specifically, we present the vertex amalgamation of m copies of the cycle Cn at a fixed vertex v∈V(Cn), for n≥6 and k2−3k+42≤m≤k2−k2, k≥3.http://www.sciencedirect.com/science/article/pii/S2405844022008842AmalgamationCycle graphPartition dimension |
| spellingShingle | Hasmawati Nurdin Hinding Budi Nurwahyu Ahmad Syukur Daming Amir Kamal Amir The partition dimension of the vertex amalgamation of some cycles Heliyon Amalgamation Cycle graph Partition dimension |
| title | The partition dimension of the vertex amalgamation of some cycles |
| title_full | The partition dimension of the vertex amalgamation of some cycles |
| title_fullStr | The partition dimension of the vertex amalgamation of some cycles |
| title_full_unstemmed | The partition dimension of the vertex amalgamation of some cycles |
| title_short | The partition dimension of the vertex amalgamation of some cycles |
| title_sort | partition dimension of the vertex amalgamation of some cycles |
| topic | Amalgamation Cycle graph Partition dimension |
| url | http://www.sciencedirect.com/science/article/pii/S2405844022008842 |
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