Abstract
The generalized Fibonacci cube $Q_h(f)$ is the graph obtained from the $h$-cube $Q_h$ by removing all vertices that contain a given binary string $f$ as a substring. In particular, the vertex set of the 3rd order generalized Fibonacci cube $Q_h(111)$ is the set of all binary strings $b_1b_2 \ldots b_h$ containing no three consecutive 1's. We present a new characterization of the 3rd order generalized Fibonacci cubes based on their recursive structure. The characterization is the basis for an algorithm which recognizes these graphs in linear time.
Highlights
The study of interconnection topologies is not just an important subject in the area of parallel or distributed systems, it initiates the research work on several new interesting classes of graphs Liu and Hsu (1992)
The vertex set of Γh is the set of all binary strings b1b2 . . . bh containing no two consecutive 1’s
A new characterization of the 3rd order generalized Fibonacci cubes is given. This characterization is the basis for the algorithm presented in the last section, which recognizes these graphs in linear time
Summary
The study of interconnection topologies is not just an important subject in the area of parallel or distributed systems, it initiates the research work on several new interesting classes of graphs Liu and Hsu (1992). The subclass of generalized Fibonacci cubes, the graphs Qh(1k), have been introduced already in Hsu and Chung (1993) and further studied in Liu et al (1994); Wasserman and Ghozati (2003); Zagaglia Salvi (1996) They are called the k-th order generalized Fibonacci cubes (of dimension h) in this paper (note that they were defined as ”generalized Fibonacci cubes” in Hsu and Chung (1993)). Hsu and Chung raised the following question Hsu and Chung (1993): Question 1.1 Given a graph, how to quickly decide whether it is the k-th order generalized Fibonacci cube of dimension h?. This characterization is the basis for the algorithm presented, which recognizes these graphs in linear time This characterization is the basis for the algorithm presented in the last section, which recognizes these graphs in linear time
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