Geodesic
$V$-graphs and their relation to the problem of locating objects in a~plane
Prikladnaâ diskretnaâ matematika, no. 4 (2013), pp. 41-46

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For two congruent figures with no common interior points, the locations in a plane are studied. A line being parallel to a shift vector intersects these pieces in two identical systems of intervals shifted by this vector. An oriented $V_n$-graph is constructed, its vertices correspond to the topologically different variants of relative position of two systems of $n$ intervals, and the edges correspond to the allowable transitions between vertices. The term of $W_n$-graph is introduced as a minimal transitive graph which contains $V_n$-graph augmented with an incident vertex. The properties of $V_n$-graphs and $W_n$-graphs are proved.
Keywords: placement of figures in a plane oriented graph, $W$-graph, the Catalan numbers, Dyck path, system slots
Mots-clés : congruent figures. 
@article{PDM_2013_4_a3,
     author = {I. G. Velichko and A. I. Zinchenko},
     title = {$V$-graphs and their relation to the problem of locating objects in a~plane},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {41--46},
     publisher = {mathdoc},
     number = {4},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2013_4_a3/}
}
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I. G. Velichko; A. I. Zinchenko. $V$-graphs and their relation to the problem of locating objects in a~plane. Prikladnaâ diskretnaâ matematika, no. 4 (2013), pp. 41-46. http://geodesic.mathdoc.fr/item/PDM_2013_4_a3/