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Entropy 2017, 19(2), 79; doi:10.3390/e19020079

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

1
State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, 100876 Beijing, China
2
School of Computer and Information Engineering, Beijing Technology and Business University, 100048 Beijing, China
*
Author to whom correspondence should be addressed.
Academic Editors: Raúl Alcaraz Martínez and Kevin H. Knuth
Received: 18 October 2016 / Revised: 25 December 2016 / Accepted: 15 February 2017 / Published: 22 February 2017
(This article belongs to the Section Information Theory)

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 ) . View Full-Text
Keywords: canonical labeling; k-mix-neighborhood subdigraph; algorithm; adjacency matrix; mix diffusion degree sequence; entire mix diffusion degree sequences canonical labeling; k-mix-neighborhood subdigraph; algorithm; adjacency matrix; mix diffusion degree sequence; entire mix diffusion degree sequences
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Hao, J.; Gong, Y.; Wang, Y.; Tan, L.; Sun, J. Using k-Mix-Neighborhood Subdigraphs to Compute Canonical Labelings of Digraphs. Entropy 2017, 19, 79.

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