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/

[1] Z. Dziechcińska-Halamoda, Z. Majcher, J. Michael and Z. Skupień, Extremum degree sets of irregular oriented graphs and pseudodigraphs, Discuss. Math. Graph Theory 26 (2006) 317-333, doi: 10.7151/dmgt.1323.

[2] Z. Dziechcińska-Halamoda, Z. Majcher, J. Michael and Z. Skupień, Large minimal irregular digraphs, Opuscula Mathematica 23 (2003) 21-24.

[3] M. Gargano, J.W. Kennedy and L.V. Quintas, Irregular digraphs, Congr. Numer. 72 (1990) 223-231.

[4] J. Górska, Z. Skupień, Z. Majcher and J. Michael, A smallest irregular oriented graph containing a given diregular one, Discrete Math. 286 (2004) 79-88, doi: 10.1016/j.disc.2003.11.049.

[5] Z. Majcher, J. Michael, J. Górska and Z. Skupień, The minimum size of fully irregular oriented graphs, Discrete Math. 236 (2001) 263-272, doi: 10.1016/S0012-365X(00)00446-5.