Geodesic
T-irreducible extensions for starlike trees
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 3, pp. 330-339
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We deal with a sort of optimal extensions of graphs, so called T-irreducible extensions. T-irreducible extension of a graph $G$ is an extension of $G$ obtained by removing a maximal set of edges from the trivial extension of $G$. A difficult starlike tree is a starlike tree that has at least one difficult node. T-irreducible extensions for nondifficult starlike trees were constructed by M. B. Abrosimov, T-irreducible extensions for palms (one of subclasses of starlike trees) were constructed by S. G. Kurnosova. Counterexamples were found to a method of Harary and Khurum, who tried to construct possible T-irreducible extensions for starlike trees. T-irreducible extensions for difficult starlike trees are constructed. 
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D. Yu. Osipov. T-irreducible extensions for starlike trees. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 3, pp. 330-339. http://geodesic.mathdoc.fr/item/ISU_2015_15_3_a11/

[1] Bogomolov A. M., Salii V. N., Algebraic foundations of the theory of discrete systems, Nauka, M., 2009 (in Russian) | MR

[2] Kurnosova S. G., “T-irreducible extensions for some classes graphs”, Theoretical Problems of Informatics and its applications, 6, Saratov Univ. Press, Saratov, 2004, 113–125 (in Russian)

[3] Harary F., Khurum M., “One node fault tolerance for caterpillars and starlike trees”, Internet J. Comput. Math., 6 (1995), 135–143 | DOI

[4] Osipov D. Yu., “On a Counterexample for a T-irreducible Extensions of Starlike Trees”, Applied Discrete Mathematics, 2014, no. 3(25), 98–102 (in Russian)

[5] Abrosimov M. B., Graph models of fault tolerance, Saratov Univ. Press, Saratov, 2012, 192 pp. (in Russian)