Abstract
Let X = (V, E) be a connected graph. Call X super restricted edge connected in short, sup-λ′, if F is a minimum edge set of X such that X − F is disconnected and every component of X − F has at least two vertices, then F is the set of edges adjacent to a certain edge with minimum edge degree in X. A bipartite graph is said to be half vertex transitive if its automorphism group is transitive on the sets of its bipartition. In this article, we show that every connected half vertex transitive graph X with n = |V(X)| ≥ 4 and \({X \ncong K_{1,n-1}}\) is λ′-optimal. By studying the λ′-superatoms of X, we characterize sup-λ′ connected half vertex transitive graphs. As a corollary, sup-λ′ connected Bi-Cayley graphs are also characterized.
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References
Ball M.O.: Complexity of network reliability computation. Networks 10, 153–165 (1980)
Bauer D., Boesch F., Suffel C., Tindell R.: Combinatorial optimization problems in the analysis and design of probabilistic networks. Networks 15, 257–271 (1985)
Bondy J.A., Murty U.S.R.: Graph Theory with Applications. Macmillan, London (1976)
Colbourn C.J.: The Combinatorics of Network Reliability. Oxford University Press, New York (1987)
Esfahanian A.H.: Generalized measures of fault tolerance with application to n-cube Networks. IEEE Trans. Comput. 38(11), 1586–1591 (1989)
Esfahanian A.H., Hakimi S.L.: On computing a conditional edge-connectivity of a graph. Inf. Process. Lett. 27, 195–199 (1988)
Harary F.: Conditional connectivity. Networks 13, 347–357 (1983)
Li Q.L., Li Q.: Reliability analysis of circulants. Networks 31, 61–65 (1998)
Liang, X.D., Meng, J.X.: Connectivity of Connected Bipartite Graphs with Two Orbits, Lecture notes in computer science, vol. 4489, pp. 334–337. Springer, Heidelberg (2007)
Mader W.: Minimale n-fach kantenzusammenhängenden Graphen. Math. Ann. 191, 21–28 (1971)
Meng J.X.: Optimally super-edge-connected transitive graphs. Discret. Math. 260, 239–248 (2003)
Meng J.X., Ji Y.H.: On a kind of restricted edge connectivity of graphs. Discret. Appl. Math. 117, 183–193 (2002)
Provan J.S., Ball M.O.: The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. Comput. 12, 777–788 (1983)
Tindell R.: Connectivity of Cayley digraphs. In: Du, D.Z., Hsu, D.F. (eds) Combinatorial Network Theory, pp. 41–46. Klumer, Dordrecht (1996)
Wang Y.Q.: Super restricted edge-connectivity of vertex-transitive graphs. Discret. Math. 289, 199–205 (2004)
Wang M., Li Q.: Conditional edge connectivity properties, reliability comparison and transitivity of graphs. Discret. Math. 258, 205–214 (2002)
Xu J.M., Xu K.L.: On restricted edge connectivity of graphs. Discret. Math. 243, 291–298 (2002)
Xu M.Y., Huang J.H., Li H.L., Li S.R.: Introduction to Group Theory (in Chinese), pp. 379–386. Academic Publishes, New York (1999)
Zhang Z., Meng J.X.: Restricted edge connectivity of edge transitive graphs. Ars Combin. LXXVIII, 297–308 (2006)
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The research is supported by NSFC (No.10671165) and NSFXJ (No. 2010211A06).
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Tian, Y., Meng, J. & Liang, X. On Super Restricted Edge Connectivity of Half Vertex Transitive Graphs. Graphs and Combinatorics 28, 287–296 (2012). https://doi.org/10.1007/s00373-011-1035-0
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DOI: https://doi.org/10.1007/s00373-011-1035-0