Abstract
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 M a x Q ( G ) . Further, the other two theorems stated below deal with identifying the second nodes added into M a x Q ( G ) . When computing C m a x ( G ) , if M a x Q ( G ) already contains the first i vertices u 1 , u 2 , ⋯ , u i , Diffusion Theorem provides a guideline on how to choose the subsequent node of M a x Q ( G ) . Besides, the Mix Diffusion Theorem shows that the selection of the ( i + 1 ) th vertex of M a x Q ( G ) for computing C m a x ( G ) is from the open mix-neighborhood subdigraph N + + ( Q ) of the nodes set Q = { u 1 , u 2 , ⋯ , u i } . It also offers two theorems to calculate the C m a x ( G ) of the disconnected digraphs. The four algorithms implemented in it illustrate how to calculate M a x Q ( 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 m a x ( G − v ) ⩽ C m a x ( G − w ) for each w ∈ S + ( G ) ∧ w ≠ v , then u 1 = v for M a x Q ( G ) .
Highlights
A canonical labeling [1,2,3] of a graph, called a canonical form [4], a canonical code [5], or an optimum code [6], is a unique string corresponding to the graph and is lexicographically smallest or largest according to the different definitions used in the studies
The computation of the canonical labeling as the digraph isomorphism problem remains an unsolved problem in computational complexity theory in the sense that no polynomial-time algorithm exists for calculating the canonical labeling of a digraph
We examine how to compute the maximum element Cmax(G) of a digraph G
Summary
A canonical labeling [1,2,3] of a graph, called a canonical form [4], a canonical code [5], or an optimum code [6], is a unique string corresponding to the graph and is lexicographically smallest or largest according to the different definitions used in the studies. Nauty is more efficient than Ullmann [19] It has almost become the industry standard used to calculate the canonical label, as well as the automorphism group. For computing the canonical labeling and automorphism group, Nauty and [20] use the depth-first search to traverse the potential intermediate nodes in the search tree. A few algorithms give the same definition of the canonical labeling as described in Definition 7, their primary purpose is not to study how to construct a canonical labeling of a digraph, but for other intentions such as to mine the frequent subdigraphs As a result, they can only work for some restricted undirected graph classes.
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