69 research outputs found

    Fault diagnosability of regular graphs

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    An interconnection network\u27s diagnosability is an important measure of its self-diagnostic capability. In 2012, Peng et al. proposed a measure for fault diagnosis of the network, namely, the hh-good-neighbor conditional diagnosability, which requires that every fault-free node has at least hh fault-free neighbors. There are two well-known diagnostic models, PMC model and MM* model. The {\it hh-good-neighbor diagnosability} under the PMC (resp. MM*) model of a graph GG, denoted by thPMC(G)t_h^{PMC}(G) (resp. thMMβˆ—(G)t_h^{MM^*}(G)), is the maximum value of tt such that GG is hh-good-neighbor tt-diagnosable under the PMC (resp. MM*) model. In this paper, we study the 22-good-neighbor diagnosability of some general kk-regular kk-connected graphs GG under the PMC model and the MM* model. The main result t2PMC(G)=t2MMβˆ—(G)=g(kβˆ’1)βˆ’1t_2^{PMC}(G)=t_2^{MM^*}(G)=g(k-1)-1 with some acceptable conditions is obtained, where gg is the girth of GG. Furthermore, the following new results under the two models are obtained: t2PMC(HSn)=t2MMβˆ—(HSn)=4nβˆ’5t_2^{PMC}(HS_n)=t_2^{MM^*}(HS_n)=4n-5 for the hierarchical star network HSnHS_n, t2PMC(Sn2)=t2MMβˆ—(Sn2)=6nβˆ’13t_2^{PMC}(S_n^2)=t_2^{MM^*}(S_n^2)=6n-13 for the split-star networks Sn2S_n^2 and t2PMC(Ξ“n(Ξ”))=t2MMβˆ—(Ξ“n(Ξ”))=6nβˆ’16t_2^{PMC}(\Gamma_{n}(\Delta))=t_2^{MM^*}(\Gamma_{n}(\Delta))=6n-16 for the Cayley graph generated by the 22-tree Ξ“n(Ξ”)\Gamma_{n}(\Delta)

    Extra Connectivity of Strong Product of Graphs

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    The gg-extraextra connectivityconnectivity ΞΊg(G)\kappa_{g}(G) of a connected graph GG is the minimum cardinality of a set of vertices, if it exists, whose deletion makes GG disconnected and leaves each remaining component with more than gg vertices, where gg is a non-negative integer. The strongstrong productproduct G1⊠G2G_1 \boxtimes G_2 of graphs G1G_1 and G2G_2 is the graph with vertex set V(G1⊠G2)=V(G1)Γ—V(G2)V(G_1 \boxtimes G_2)=V(G_1)\times V(G_2), where two distinct vertices (x1,y1),(x2,y2)∈V(G1)Γ—V(G2)(x_{1}, y_{1}),(x_{2}, y_{2}) \in V(G_1)\times V(G_2) are adjacent in G1⊠G2G_1 \boxtimes G_2 if and only if x1=x2x_{1}=x_{2} and y1y2∈E(G2)y_{1} y_{2} \in E(G_2) or y1=y2y_{1}=y_{2} and x1x2∈E(G1)x_{1} x_{2} \in E(G_1) or x1x2∈E(G1)x_{1} x_{2} \in E(G_1) and y1y2∈E(G2)y_{1} y_{2} \in E(G_2). In this paper, we give the gΒ (≀3)g\ (\leq 3)-extraextra connectivityconnectivity of G1⊠G2G_1\boxtimes G_2, where GiG_i is a maximally connected kiΒ (β‰₯2)k_i\ (\geq 2)-regular graph for i=1,2i=1,2. As a byproduct, we get gΒ (≀3)g\ (\leq 3)-extraextra conditional fault-diagnosability of G1⊠G2G_1\boxtimes G_2 under PMCPMC model
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