Abstract
Connectivity has been used in the past to describe the stability of graphs. If two graphs have the same connectivity, then it does not distinguish between these graphs. That is, the connectivity is not a good measure of graph stability. Then we need other graph parameters to describe the stability. Suppose that two graphs have the same connectivity and the order (the number of vertices or edges) of the largest components of these graphs are not equal. Hence, we say that these graphs must be different in respect to stability and so we can define a new measure which distinguishes these graphs. In this paper, the Weak-Integrity of a graph G is introduced as a new measure of stability in this sense and it is defined as I w (G)=min S⊂V(G){S+m e (G−S)}, where m e (G−S) denotes the number of edges of the largest component of G−S. We give the weak-integrity of graphs obtained via various operations that are unary, such as powers, and binary, such as union, composition, product and corona.
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Atıcı, M. and Kırlangıc, A.: Counterexamples to the theorems of integrity of prisms and ladders, J. Combin. Math. Combin. Comput. 34 (2000), 119–127.
Bagga, K. S. and Deogun, J. S.: A variation on the edge-integrity, Congr. Numer. 94 (1992), 207–211.
Bagga, K. S. and Deogun, J. S.: On the pure edge-integrity of graphs, In: Graph Theory, Combinatorics, and Algorithms, Vols 1, 2, Kalamazoo, MI, 1992, pp. 301–310.
Bagga, K. S., Beineke, L. W., Goddard, W. D., Lipman, M. J. and Pippert, R. E.: A survey of integrity, Discrete Appl. Math. 37/38 (1992), 13–28.
Barefoot, C. A., Entringer, R. and Swart, H. C.: Vulnerability in graphs - a comparative survey, J. Combin. Math. Combin. Comput. 1 (1987), 13–22.
Barefoot, C. A., Entringer, R. and Swart, H. C.: Integrity of trees and powers of cycles, Congr. Numer. 58 (1987), 103–114.
Choudum, A. S. and Priya, N.: Tenacity of complete graph products and grids, Networks 34 (1999), 192–196.
Cozzens, M. B.: Stability measures and data fusion networks, Graph Theory Notes New York XXVI (1994), 8–14.
Cozzens, M. B., Moazzami, D. and Stueckle, S.: The tenacity of a graph, In: Proc. Seventh Internat. Conf. Theory and Applications of Graphs, Wiley, New York, 1995, pp. 1111–1122.
Harary, F.: Graph Theory, Addison-Wesley, New York, 1972.
Kırlangıc., A.: The edge-integrity of some graphs, J. Combin. Math. Combin. Comput. 37 (2001), 139–148.
Kırlangıc., A. and Ozan, A.: The neighbour-integrity of total graphs, Internat. J. Comput. Math. 76(1) (2000), 25–33.
Piazza, B. L., Roberts, S. F. and Stueckle, K. S.: Edge-tenacious networks, Networks 25 (1995), 7–17.
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Kirlangic, A. On the Weak-Integrity of Graphs. Journal of Mathematical Modelling and Algorithms 2, 81–95 (2003). https://doi.org/10.1023/A:1024945012302
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DOI: https://doi.org/10.1023/A:1024945012302