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

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 di f f usion outdegree sequence and entire mix di f f usion 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 u 1 , u 2 , · · · , 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 = {u 1 , u 2 , · · · , u i }. It also offers two theorems to calculate the C max (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 u 1 = v for MaxQ(G).

Use the K-Neighborhood Subgraphs to Compute Canonical Labelings of Graphs

“…Observe that a side of G corresponds to 1 bit of the upper triangular part of the adjacency matrix A(G). To maximize C(G) by maximizing C 2 (G), one must make u 2 belong to N(u 1 ) such that makes a 1,2 = 1 (see (1)). Otherwise, if u 2 / ∈ N(u 1 ), then a 1,2 = 0 (see (1)) and C(G) =C max (G).…”

“…Proof. (1). To maximize C(G) by maximizing C 2 (G) (see (1)), it follows that u 2 must satisfy condition…”

“…This paper is the close companion to Reference [1], in which a novel theory and method are presented for calculating the canonical labelings of digraphs. The center subgraph Cen(G) [2] can also be used to determine the first vertex u 1 added into MaxQ(G) of undirected graphs.…”