Abstract
We introduce a new class of graphs called diametral path graphs that properly contains the class of asteroidal triple-free graphs and the class of dominating pair graphs. We characterize the trees as well as the chordal graphs that are diametral path graphs. We present an O(n 3 m) algorithm deciding whether a given graph has a dominating diametral path. Finally, we study the structure of minimum connected dominating sets in diametral path graphs.
This research was supported in part by the Office of Naval Research under Grant No. N0014-91-J-1693.
Part of this research was done while the second author was at IRISA Rennes (France) supported by CHM Fellowship.
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References
K.S. Bagga, L.W. Beineke, W.D. Goddard, M.J. Lipman, and R.E. Pippert, A survey of integrity, to appear in Discrete Applied Mathematics.
C.A. Barefoot, R. Entringer, and H. Swart, Vulnerability in graphs — A comparative survey, J. Combin. Math. Combin. Comput. 1 (1987), 12–22.
J.C. Bermond, and B. Bollobás, The diameter of graphs — a survey, Congressus Numerantium 81 (1981), 3–27.
J.C. Bermond, J. Bond, M. Paoli, and C. Peyrat, Graphs and interconnection networks: diameter and vulnerability, Surveys in Combinatorics, London Math. Soc. Lecture Notes 82 (1983), 1–30.
J.A. Bondy, U.S.R. Murty, Graph theory with applications, Macmilliam Press Ltd., 1976.
A. BrandstÄdt, Special graph classes — a survey, Schriftenreihe des Fachbereichs Mathematik, SM-DU-199, UniversitÄt Duisburg, 1991.
D.G. Corneil, S. Olariu, and L. Stewart, Asteroidal triple-free graphs, Proceedings of the 19th International Workshop on Graph-Theoretic Concepts in Computer Science WG'93, Lecture Notes in Computer Science, Vol. 790, Springer-Verlag, 1994, pp. 211–224.
D.G. Corneil, S. Olariu, and L. Stewart, Asteroidal triple-free graphs, Manuscript, 1994, (revised version of [7]).
D.G. Corneil, S. Olariu, and L. Stewart, A linear time algorithm to compute a dominating path in an AT-free graph, Information Processing Letters, 54 (1995), 253–258.
C.J. Colbourn, and L.K. Stewart, Permutation graphs: connected domination and steiner trees, Discrete Mathematics 86 (1991), 179–190.
J.S. Deogun, Diametral path graphs, presented at the Second International Conference On Graph Theory and Algorithms, San Francisco, August 1989.
M.C. Golumbic, Algorithmic graph theory and perfect graphs, Academic Press, New York, 1980.
D. Kratsch, P. Damaschke, and A. Lubiw, Dominating cliques in chordal graphs, Discrete Mathematics 128 (1994), 269–276.
D. Kratsch, L. Stewart, Domination on cocomparability graphs, SIAM Journal on Discrete Mathematics 6 (1993), 400–417.
R. Ogier, and N. Shacham, A distributed algorithm for finding shortest pairs of disjoint paths, IEEE INFOCOM'89 1 (1989), 173–182.
O. Ore, Diameters in graphs, J. Combin. Theory 5 (1968), 75–81.
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© 1995 Springer-Verlag Berlin Heidelberg
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Deogun, J.S., Kratsch, D. (1995). Diametral path graphs. In: Nagl, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 1995. Lecture Notes in Computer Science, vol 1017. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60618-1_87
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DOI: https://doi.org/10.1007/3-540-60618-1_87
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