https://ogma.newcastle.edu.au/vital/access/ /manager/Index ${session.getAttribute("locale")} 5 The g-good-neighbor and g-extra diagnosability of networks https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:37827 G = (V, E). In 2012, a measurement for fault tolerance of the graph was proposed by Peng et al. This measurement is called the g-good-neighbor diagnosability that restrains every fault-free node to contain at least g fault-free neighbors. In 2016, Zhang et al. proposed a new measurement for fault diagnosis of the graph, namely, the g-extra diagnosabil-ity, which restrains that every fault-free component has at least (g+1) fault-free nodes. A fault set F ⊆ V is called a g-good-neighbor faulty set if the degree d(v)g for every vertex v in G.−F. A g-good-neighbor cut of G is a g-good-neighbor faulty set F such that GF is disconnected. The minimum cardinality of g-good-neighbor cuts is said to be the g-good-neighbor connectivity of G. A fault set F ⊆ V is called a g-extra faulty set if every component of GF has at least (g+1) vertices. A g-extra cut of G is a g-extra faulty set F such that GF is disconnected. The minimum cardinality of g-extra cuts is said to be the g-extra connectivity of G. The g-good-neighbor (extra) diagnosability and g-good-neighbor (extra) connectivity of many well-known graphs have been widely investigated. In this paper, we show the relationship between the g-good-neighbor (extra) diagnosability and g-good-neighbor (extra) connectivity of graphs.]]> Wed 12 May 2021 09:53:36 AEST ]]> The 2-good-neighbor connectivity and 2-good-neighbor diagnosability of bubble-sort star graph networks https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:32182 n has many good properties. In this paper, we prove that 2-good-neighbor connectivity of BSn is 8n - 22 for n ≥ 5 and the 2-good-neighbor connectivity of BS₄ is 8; the 2-good-neighbor diagnosability of BSn is 8n - 19 under the PMC model and MM* model for n ≥ 5.]]> Wed 09 May 2018 15:40:06 AEST ]]> The connectivity and nature diagnosability of expanded k-ary n-cubes https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:32179 kn has many good properties. In this paper, we prove that (1) the connectivity of XQkn is 4n; (2) the nature connectivity of XQkn is 8n − 4; (3) the nature diagnosability of XQkn under the PMC model and MM∗ model is 8n − 3 for n ≥ 2.]]> Wed 09 May 2018 14:58:09 AEST ]]> Sufficient conditions for graphs to be maximally 4-restricted edge connected https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:33000 k(G), is defined as the cardinality of a minimum k-restricted edge cut. A connected graph G is said to be λk-connected if G has a k-restricted edge cut. Let ξk(G) = min{|[X, ̅X ]| : |X| = k, G[X] is connected}, where ̅X = V (G)X. A graph G is said to be maximally k-restricted edge connected if λk(G) = ξk(G). In this paper we show that if G is a λ₄-connected graph with λ₄(G) ≤ ξ₄(G) and the girth satisfies g(G) ≥ 8, and there do not exist six vertices u₁, u₂, u₃, v₁, v₂ and v₃ in G such that the distance d(ui, vj) ≥ 3, (1 ≤ i, j ≤ 3), then G is maximally 4-restricted edge connected.]]> Tue 03 Sep 2019 18:17:58 AEST ]]> The nature diagnosability of bubble-sort star graphs under the PMC model and MM* model https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:32057 n-dimensional bubble-sort star graph BSn has many good properties. In this paper, we prove that the nature diagnosability of BSn is 4n - 7 under the PMC model for n ≥ 4, the nature diagnosability of BSn is n ≥ 4 under the MM* model for n ≥ 5 .]]> Thu 26 Apr 2018 14:24:55 AEST ]]> The 2-good-neighbor diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM* model https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:24872 n generated by the transposition tree Γn has many good properties. In this paper, we give that the 2-good-neighbor diagnosability of CΓn under the PMC model and MM⁎ model is g(n−2)−1, where n≥4and g is the girth of CΓn.]]> Sat 24 Mar 2018 07:11:20 AEDT ]]> The tightly super 3-extra connectivity and diagnosability of locally twisted cubes https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:32178 et al. proposed the g-extra diagnosability of G, which restrains that every component of G - S has at least (g + 1) vertices. The locally twisted cube LTQn is applied widely. In this paper, we show that LTQn is tightly (4n-9) super 3-extra connected for n≥6 and the 3-extra diagnosability of LTQn under the PMC model and MM* model is 4n - 6 for n≥5 and n≥7, respectively.]]> Mon 23 Sep 2019 12:57:15 AEST ]]> An algorithm for the orientation of complete bipartite graphs https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:32495 G be a graph with vertex set V(G) and edge set E(G). We consider the problem of orienting the edges of a complete bipartite graph Kn,n so only two different in-degrees a and b occur. An obvious necessary condition for orienting the edges of G so that only two in-degrees a and b occur, is that there exist positive integers s and t satisfying s+t=|V(G)| and as+bt=|V(G)|. In this paper, we show that the necessary condition is also sufficient for a complete bipartite graph Kn,n. Furthermore, we give the algorithms of orientations with only two in-degrees of Kn,n.]]> Mon 23 Sep 2019 11:31:42 AEST ]]> The 1-good-neighbour diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM* model https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:32988 et al. proposed a new measure for fault tolerance of the system, which is called g-good-neighbour diagnosability that restrains every fault-free node containing at least g fault-free neighbours. As a favourable topology structure of interconnection networks, the Cayley graph CΓn generated by the transposition tree Γn has many good properties. In this paper, we give that the 1-good-neighbour diagnosability of CΓn under the PMC model and MM∗ model is 2n−3 except the bubble-sort graph B₄ under MM∗ model, where n≥4, and the 1-good-neighbour diagnosability of B₄ under the MM∗ model is 4.]]> Fri 17 Aug 2018 15:44:20 AEST ]]> g-good-neighbor conditional diagnosability of star graph networks under PMC model and MM* model https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:32184 n has many good properties. In this paper, we establish the g-good-neighbor conditional diagnosability of Sn under the PMC model and MM* model.]]> Fri 11 May 2018 13:24:22 AEST ]]>