On critical and maximal digraphs

07/24/2018 ∙ by G. Š. Fridman, et al. ∙ 0 ∙ share

This paper is devoted to the study of directed graphs with extremal properties relative to certain metric functionals. We characterize up to isomorphism critical digraphs with infinite values of diameter, quasi-diameter, radius and quasi-radius. Moreover, maximal digraphs with finite values of radius and quasi-diameter are studied.

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References

[1] Aigner M. (1979) Combinatorial theory. Springer-Verlag, Berlin.

[2] Fridman G. Š. (1971) Diametrically-critical graphs of infinite diameter. In: On some questions of theoretical cybernetics and programming algorithms, G.P.Bagrinovski and A.A.Ljapunov, eds. Inst. of Hydrodinamics SO AN SSSR, Novosibirsk, 36-43 (in Russian).

[3] Fridman G. Š. (1973) On radius critical directed graphs. Dokl. Akad. Nauk SSSR 212, No. 3, 565-568 (in Russian); Engl. transl. Soviet Math. Dokl. 14, No. 5, 1435-1439.

[4] Fridman G. Š. (1977) Maximal directed graphs of finite quasi-diameter. In: Abstracts of the third accounting scientific-methodical conference. Omsk State University, Omsk, 98-99 (in Russian).

[5] Gliviak F. (1975) On radially critical graphs. In Recent Advances in Graph Theory. Proc. Sympos. Prague 1974. Academica, Prague, 207-221.

[6] Gliviak F., Knor M., Šoltés L. (1994) On radially maximal graphs. Australasian J. of Comb. 9, 275-284.

[7] Gliviak F., Knor M. (1995) On radially extremal digraphs. Mathematica Bohemica. 120, No. 1, 41-55.

[8] Ismailov Š. M. (1970) On the connection of the radius of a digraph with the number of its paths. Upravl. sistemy. Nauka, Novosibirsk, No. 4-5, 35-38 (in Russian).

[9] Ismailov Š. M. (1971) The number of arcs of a digraph of a given radius with a given number of vertices and strong components. Dokl. Akad. Nauk Azerbaidžan SSR 27, No. 2, 8-12 (in Russian).

[10] Ismailov Š. M. (1973) An upper bound for the number of arcs of a nonbiconnected digraph with a given number of bicomponents and radius. VINITI manuscript, No. 3245-71 Dep. (in Russian).

[11] Mel’nikov L. S. (1970) Critical digraphs with given diameter. Upravl. sistemy. Nauka, Novosibirsk, No. 7, 37-45 (in Russian).

[12] Ore O. (1968) Diameters in graphs. J. combin theory 5, 75-81.

[13] Vizing V. G. (1967) On the number of edges in a graph with given radius. Dokl. Akad. Nauk SSSR 173, 1245-1246 (in Russian); Engl. transl. Soviet Math. Dokl. 8, 535-536.

[14] Zykov A. A. (1969) Theory of finite graphs. I. Nauka, Novosibirsk (in Russian).

On critical and maximal digraphs

Authors

Revisiting Radius, Diameter, and all Eccentricity Computation in Graphs through Certificates

On the Size of Levenshtein Balls

There is no lattice tiling of Z^n by Lee spheres of radius 2 for n≥ 3

Density of Binary Disc Packings: Playing with Stoichiometry

Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain

Alternation diameter of a product object

A Linear-Time Algorithm for Radius-Optimally Augmenting Paths in a Metric Space

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References