Geodesic
$\mathrm{T}$-irreducible extensions for unions of complete graphs
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 5 (2005) no. 1, pp. 107-115.

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$\mathrm{T}$-irreducible extension is оne of kinds of optimal extensions of graphs. Constructions of optimal extensions are used in diagnosis of discrete systems аnd in cryptography. A graph $H$ with $n+1$ vertices is called аn extension of а graph $G$ with $n$ vertices if $G$ can be embedded in every maximal subgraph of $H$. Any graph $G$ has the trivial extension that is the join $G+v$ of $G$ with some outer vertex $v$. $\mathrm{T}$-irreducible extensions are obtained from the trivial extension bу removal of maximal number of edges in such а way that the extension property is preserved. The problem of finding of the initial graph from аnу of its $\mathrm{T}$-irreducible extensions has а linear complexity, but until nоw there is nо efficient algorithm for finding of all $\mathrm{T}$-irreducible extensions of а given graph. Graphs studied in this paper are unions of complete graphs each of which has more than оne vertex. An algorithm of construction of all $\mathrm{T}$-irreducible extensions for such graphs is presented. Also an estimate of а total amount of the resultihg extensions is made. 
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S. G. Kurnosova. $\mathrm{T}$-irreducible extensions for unions of complete graphs. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 5 (2005) no. 1, pp. 107-115. http://geodesic.mathdoc.fr/item/ISU_2005_5_1_a10/

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