Geodesic
A Sokoban-type game and arc deletion within irregular digraphs of all sizes
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 3, pp. 611-622

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Digraphs in which ordered pairs of out- and in-degrees of vertices are mutually distinct are called irregular, see Gargano et al. [3]. Our investigations focus on the problem: what are possible sizes of irregular digraphs (oriented graphs) for a given order n? We show that those sizes in both cases make up integer intervals. The extremal sizes (the endpoints of these intervals) are found in [1,5]. In this paper we construct, with help of Sokoban-type game, n-vertex irregular oriented graphs (irregular digraphs) of all intermediate sizes.
Keywords: irregular digraph, all sizes 
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Dziechcińska-Halamoda, Zyta; Majcher, Zofia; Majcher, Zofia; Skupień, Zdzisław. A Sokoban-type game and arc deletion within irregular digraphs of all sizes. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 3, pp. 611-622. http://geodesic.mathdoc.fr/item/DMGT_2007_27_3_a18/