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. 
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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/

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