Geodesic
Graph congruences: some combinatorial properties
Prikladnaya Diskretnaya Matematika. Supplement, no. 5 (2012), pp. 93-94 
E. O. Karmanova. Graph congruences: some combinatorial properties. Prikladnaya Diskretnaya Matematika. Supplement, no. 5 (2012), pp. 93-94. http://geodesic.mathdoc.fr/item/PDMA_2012_5_a47/
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     author = {E. O. Karmanova},
     title = {Graph congruences: some combinatorial properties},
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     year = {2012},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2012_5_a47/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

A congruence relation of a path is an equivalence relation on the set of its vertices all of whose classes are independent subsets. It is proved (theorem 1) that the number of all congruence relations of a path with $m$ edges equals to the number of all equivalence relations on a $m$-element set. For a given connected graph $G$ theorem 2 determines the length of the shortest path whose quotient-graph is $G$.

[1] Karmanova E. O., “O kongruentsiyakh tsepei”, Prikladnaya diskretnaya matematika, 2011, no. 2(12), 96–100

[2] Karmanova E. O., “Kongruentsii tsepei: nekotorye kombinatornye svoistva”, Prikladnaya diskretnaya matematika, 2012, no. 2(12), 86–89