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Digraphs in which ordered pairs of outand 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 612 Z. Dziechcińska-Halamoda, Z. Majcher, ... 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.

A Sokoban-type game and arc deletion within irregular digraphs of all sizes