Diameter-girth sufficient conditions for optimal extraconnectivity in graphs
For a connected graph G, the rth extraconnectivity r(G)is defined asthe minimum cardinality of a cutset X such that all remaining components after the deletion of the vertices of X have at least r + 1 vertices. The standard connectivity and superconnectivity correspond to 0(G) and 1(G), respectively...
| Autores: | , , , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2008 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/163348 |
| Acceso en línea: | https://hdl.handle.net/11441/163348 https://doi.org/10.1016/j.disc.2007.07.012 |
| Access Level: | acceso abierto |
| Palabra clave: | Connectivity Superconnectivity Cutset Diameter Girth |
| Sumario: | For a connected graph G, the rth extraconnectivity r(G)is defined asthe minimum cardinality of a cutset X such that all remaining components after the deletion of the vertices of X have at least r + 1 vertices. The standard connectivity and superconnectivity correspond to 0(G) and 1(G), respectively. The minimum r-tree degree of G, denoted by r(G), is the minimum cardinality of N(T ) taken over all trees T ⊆ G of order |V (T )| = r + 1, N(T ) being the set of vertices not in T that are neighbors of some vertex of T. When r = 1, any such considered tree is just an edge of G. Then, 1(G) is equal to the so-called minimum edge-degree of G, defined as (G) = min{d(u) + d(v) − 2 : uv ∈ E(G)}, where d(u) stands for the degree of vertex u. A graph G is said to be optimally r-extraconnected, for short r-optimal, if r(G) r(G). In this paper, we present some sufficient conditions that guarantee r(G) r(G) for r 2. These results improve some previous related ones, and can be seen as a complement of some others which were obtained by the authors for r = 1. |
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