On the g-good-neighbor connectivity of graphs

Abstract

Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network G. In 1996, Fàbrega and Fiol proposed the g-good-neighbor connectivity of G. In this paper, we show that 1 ≤ κ g (G) ≤ n − 2g − 2 for 0 ≤ g ≤ ∆(G), n−3 2 , and graphs with κ g (G) = 1, 2 and trees with κ g (T n ) = n − t for 4 ≤ t ≤ n+2 2 are characterized, respectively. In the end, we get the three extremal results for the g-good-neighbor connectivity. The g-extra (edge-)connectivityFàbrega and Fiol [8,9] introduced the following measures. A subset of vertices S is said to be a cutset if G − S is not connected. A cutset S is called an R g -cutset, where g is a non-negative integer, if every component of G − S has at least g + 1 vertices. If G has at least one R g -cutset, the g-extra connectivity of G, denoted by κ g (G), is then defined as the minimum cardinality over all R g -cutsets of G. A connected graph G is said to be g-extra connected if G has a g-extra cut.Given a graph and a non-negative integer g, the g-extra edge-connectivity, written as λ g (G), is the minimal cardinality of a set of edges in G, if it exists, whose deletion disconnects G and leaves each remaining component with more than g vertices.Note that κ 0 (G) = κ(G) and λ 0 (G) = λ(G) for any graph if is not a complete graph. Therefore, the g-extra connectivity (resp. g-extra edge connectivity) can be regarded as a generalization of the classical connectivity (resp. classical edge connectivity) that provides measures that are more accurate for reliability and fault tolerance for large-scale parallel processing systems. Regarding the computational complexity of the problem, based on thorough research, no polynomial-time algorithm has been presented to compute κ g for a general graph; nor has there been any tight upper bound for κ g [6]. However, λ 1 can be computed by solving numerous network flow problems [7]. The g-good-neighbor connectivityLet G = (V, E) be connected. A fault set F ⊆ V is called a g-good-neighbor faulty set if |N (v)∩(V −F )| ≥ g for every vertex v in V − F . A g-good-neighbor cut of G is a g-good-neighbor faulty set F such that G − F is disconnected. The minimum cardinality of g-good-neighbor cuts is said to be the ggood-neighbor connectivity of G, denoted by κ g (G). A connected graph G is said to be g-good-neighbor

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