AI-enabled blockchain refers to the use of AI to enable the analysis and decision-making processes based on data collected, shared, and stored by blockchain. This helps overcome some of the existing challenges in blockchain applications. Despite the growing number of review papers on blockchain and AI, there is a dearth of literature on AI-enabled blockchain in business scenarios. This study uses bibliometric-content analysis to (1) identify three stages of development of AI-enabled blockchain literature and point out the increasing diversity of technological applications; (2) identify the strongest foci of extant literature; (3) unveil the roles of AI-enabled blockchain in 10 application sectors, and identify the key roles of AI in enabling blockchain applications; (4) conclude the referred ethical issues from three levels and make further discussion. The findings present the trends of AI-enabled blockchain and could help developers and service providers better manage the use and ethical issues of AI in blockchain applications.

A well-studied geometric object in combinatorial optimization is the perfect matching polytope of a graph $G$---the convex hull of the incidence vectors of all perfect matchings of $G$. In any investigation concerning the perfect matching polytope, one may assume that $G$ is matching covered---that is, $G$ is a connected graph (of order at least two) and each edge of $G$ lies in some perfect matching. A graph $G$ is Birkhoff--von Neumann if its perfect matching polytope is characterized solely by nonnegativity and degree constraints. A result of Balas [ North Holland Math. Stud., 59 (1981), pp. 1--13] implies that $G$ is Birkhoff--von Neumann if and only if $G$ does not contain a pair of vertex-disjoint odd cycles $(C_1,C_2)$ such that $G-V(C_1)-V(C_2)$ has a perfect matching. It follows immediately that the corresponding decision problem is in co-$\mathcal{NP}$. However, it is not known to be in $\mathcal{NP}$. The problem is in $\mathcal{P}$ if the input graph is planar---due to a result of Carvalho, Lucchesi, and Murty [ J. Combin. Theory Ser. B, 92 (2004), pp. 319--324]. More recently, these authors, along with Kothari [ SIAM J. Discrete Math., 32 (2018), pp. 1478--1504], have shown that this problem is equivalent to the seemingly unrelated problem of deciding whether a given graph is $\overline{C_6}$-free. The combinatorial diameter of a polytope is the diameter of its $1$-skeleton graph. A graph $G$ is PM-compact if the combinatorial diameter of its perfect matching polytope equals one. Independent results of Balinski and Russakoff [ SIAM Rev., 16 (1974), pp. 516--525] and of Chvátal [ J. Combin. Theory Ser. B, 18 (1975), pp. 138--154] imply that $G$ is PM-compact if and only if $G$ does not contain a pair of vertex-disjoint even cycles $(C_1,C_2)$ such that $G-V(C_1)-V(C_2)$ has a perfect matching. Once again the corresponding decision problem is in co-$\mathcal{NP}$, but it is not known to be in $\mathcal{NP}$. The problem is in $\mathcal{P}$ if the input graph is bipartite or is near-bipartite---due to a result of Wang et al. [ Discrete Math., 313 (2013), pp. 772--783]. In this paper, we consider the “intersection” of the aforementioned problems. We give an alternative description of matching covered graphs that are Birkhoff--von Neumann as well as PM-compact; our description implies that the corresponding decision problem is in $\mathcal{P}$.

The obnoxious center problem in a graph G asks for a location on an edge of the graph such that the minimum weighted distance from this point to a vertex of the graph is as large as possible. We derive algorithms with linear running time for the cases when G is a path or a star, thus improving previous results of Tamir [SIAMJ. Discrete Math, 1 (1988), pp. 377--396]. For subdivided stars we present an algorithm of running time O(n log n). For general trees, we improve an algorithm of Tamir [SIAM J. Discrete Math, 1 (1988), pp. 377--396] by a factor of log n. Moreover, a linear algorithm for the unweighted center problem on an arbitrary tree with neutral and obnoxious vertices is described.

For the bandwidth B(G) and the cyclic bandwidth Bc(G) of a graph G, it is known that ½B(G) ≤ Bc(G) ≤ B(G). In this paper, the criterion conditions for two extreme cases Bc(G) = B(G) and Bc(G) = ½B(G) are studied. From this, some exact values of Bc(G) for special graphs can be obtained. © 1997 John Wiley & Sons, Inc. Networks 29: 135–140, 1997