Sufficient conditions for lambda '-optimality of graphs with small conditional diameter
A restricted edge-cut S of a connected graph G is an edge-cut such that G − S has no isolated vertex. The restricted edgeconnectivity λ (G) is the minimum cardinality over all restricted edge-cuts. A graph is said to be λ -optimal if λ (G) = ξ(G), where ξ(G) denotes the minimum edge-degree of G defi...
| Autores: | , , , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2005 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/163836 |
| Acceso en línea: | https://hdl.handle.net/11441/163836 https://doi.org/10.1016/j.ipl.2005.05.006 |
| Access Level: | acceso abierto |
| Palabra clave: | Fault tolerance Restricted edge-connectivity Conditional diameter |
| Sumario: | A restricted edge-cut S of a connected graph G is an edge-cut such that G − S has no isolated vertex. The restricted edgeconnectivity λ (G) is the minimum cardinality over all restricted edge-cuts. A graph is said to be λ -optimal if λ (G) = ξ(G), where ξ(G) denotes the minimum edge-degree of G defined as ξ(G) = min{d(u)+d(v)−2: uv ∈ E(G)}. The P-diameter of G measures how far apart a pair of subgraphs satisfying a given property P can be, and hence it generalizes the standard concept of diameter. In this paper we prove two kind of results, according to which property P is chosen. First, let D1 (resp. D2) be the P-diameter where P is the property that the corresponding subgraphs have minimum degree at least one (resp. two). We prove that a graph with odd girth g is λ -optimal if D1 g − 2 and D2 g − 5. For even girth we obtain a similar result. Second, let F ⊂ V (G) with |F| = δ − 1, δ 2, being the minimum degree of G. Using the property Q of being vertices of G − F we prove that a graph with girth g /∈ {4, 6, 8} is λ -optimal if this Q-diameter is at most 2 (g − 3)/2 . |
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