Using k-Mix-Neighborhood Subdigraphs to Compute Canonical Labelings of Digraphs.

This paper presents a novel theory and method to calculate the canonical labelings of digraphs whose definition is entirely different from the traditional definition of Nauty. It indicates the mutual relationships that exist between the canonical labeling of a digraph and the canonical labeling of its complement graph. It systematically examines the link between computing the canonical labeling of a digraph and the k-neighborhood and k-mix-neighborhood subdigraphs. To facilitate the presentation, it introduces several concepts including mix diffusion outdegree sequence and entire mix diffusion outdegree sequences. For each node in a digraph G, it assigns an attribute m_NearestNode to enhance the accuracy of calculating canonical labeling. The four theorems proved here demonstrate how to determine the first nodes added into MaxQ(G). Further, the other two theorems stated below deal with identifying the second nodes added into MaxQ(G). When computing C_max(G), if MaxQ(G) already contains the first i vertices u₁, u₂, ..., u_i, Diffusion Theorem provides a guideline on how to choose the subsequent node of MaxQ(G). Besides, the Mix Diffusion Theorem shows that the selection of the (i + 1)th vertex of MaxQ(G) for computing C_max(G) is from the open mix-neighborhood subdigraph N⁺⁺(Q) of the nodes set Q = {u₁, u₂, ..., u_i}. It also offers two theorems to calculate the Cmax(G) of the disconnected digraphs. The four algorithms implemented in it illustrate how to calculate MaxQ(G) of a digraph. Through software testing, the correctness of our algorithms is preliminarily verified. Our method can be utilized to mine the frequent subdigraph. We also guess that if there exists a vertex v ∈ S⁺(G) satisfying conditions C_max(G - v) ≤ C_max(G - w) for each w ∈ S⁺(G) ^ w ≠ v, then u₁ = v for MaxQ(G). [ABSTRACT FROM AUTHOR]