Component Order Edge Connectivity, Vertex Degrees, and Integer Partitions
Given a finite, simple graph G, the k-component order connectivity (resp. edge connectivity) of G is the minimum number of vertices (resp. edges) whose removal results in a subgraph in which every component has an order of at most k − 1. In general, determining the k-component order edge connectivit...
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| Format: | Article |
| Language: | English |
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Georgia Southern University
2025-01-01
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| Series: | Theory and Applications of Graphs |
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| Online Access: | https://digitalcommons.georgiasouthern.edu/tag/vol12/iss1/1/ |
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| author | Michael R. Yatauro |
| author_facet | Michael R. Yatauro |
| author_sort | Michael R. Yatauro |
| collection | DOAJ |
| description | Given a finite, simple graph G, the k-component order connectivity (resp. edge connectivity) of G is the minimum number of vertices (resp. edges) whose removal results in a subgraph in which every component has an order of at most k − 1. In general, determining the k-component order edge connectivity of a graph is NP-hard. We identify conditions on the vertex degrees of G that can be used to imply a lower bound on the k-component order edge connectivity of G. We will discuss the process for generating such conditions for a lower bound of 1 or 2, and we explore how the complexity increases when the desired lower bound is 3 or more. In the process, we provide new proofs of related results concerning k component order connectivity, and we prove some relevant results about integer partitions. |
| format | Article |
| id | doaj-art-becf1a0febae4f1db1547279fe76d393 |
| institution | DOAJ |
| issn | 2470-9859 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Georgia Southern University |
| record_format | Article |
| series | Theory and Applications of Graphs |
| spelling | doaj-art-becf1a0febae4f1db1547279fe76d3932025-08-20T03:06:20ZengGeorgia Southern UniversityTheory and Applications of Graphs2470-98592025-01-0112111410.20429/tag.2025.120101Component Order Edge Connectivity, Vertex Degrees, and Integer PartitionsMichael R. YatauroGiven a finite, simple graph G, the k-component order connectivity (resp. edge connectivity) of G is the minimum number of vertices (resp. edges) whose removal results in a subgraph in which every component has an order of at most k − 1. In general, determining the k-component order edge connectivity of a graph is NP-hard. We identify conditions on the vertex degrees of G that can be used to imply a lower bound on the k-component order edge connectivity of G. We will discuss the process for generating such conditions for a lower bound of 1 or 2, and we explore how the complexity increases when the desired lower bound is 3 or more. In the process, we provide new proofs of related results concerning k component order connectivity, and we prove some relevant results about integer partitions.https://digitalcommons.georgiasouthern.edu/tag/vol12/iss1/1/component order connectivitycomponent order edge connectivitydegree sequencebest monotoneinteger partition |
| spellingShingle | Michael R. Yatauro Component Order Edge Connectivity, Vertex Degrees, and Integer Partitions Theory and Applications of Graphs component order connectivity component order edge connectivity degree sequence best monotone integer partition |
| title | Component Order Edge Connectivity, Vertex Degrees, and Integer Partitions |
| title_full | Component Order Edge Connectivity, Vertex Degrees, and Integer Partitions |
| title_fullStr | Component Order Edge Connectivity, Vertex Degrees, and Integer Partitions |
| title_full_unstemmed | Component Order Edge Connectivity, Vertex Degrees, and Integer Partitions |
| title_short | Component Order Edge Connectivity, Vertex Degrees, and Integer Partitions |
| title_sort | component order edge connectivity vertex degrees and integer partitions |
| topic | component order connectivity component order edge connectivity degree sequence best monotone integer partition |
| url | https://digitalcommons.georgiasouthern.edu/tag/vol12/iss1/1/ |
| work_keys_str_mv | AT michaelryatauro componentorderedgeconnectivityvertexdegreesandintegerpartitions |