Asymptotically Effective Method to Explore Euler Path in a Graph

Euler path is one of the most interesting and widely discussed topics in graph theory. An Euler path (or Euler trail) is a path that visits every edge of a graph exactly once. Similarly, an Euler circuit (or Euler cycle) is an Euler trail that starts and ends on the same node of a graph. A graph having Euler path is called Euler graph. While tracing Euler graph, one may halt at arbitrary nodes while some of its edges left unvisited. In this paper, we have proposed some precautionary steps that should be considered in exploring a deadlock-free Euler path, i.e., without being halted at any node. Simulation results show that our proposed approach improves the process of exploring the Euler path in an undirected connected graph without interruption. Furthermore, our proposed algorithm is complete for all types of undirected Eulerian graphs. The paper concludes with the proofs of the correctness of proposed algorithm and its computation complexity.

Full-Custom Design and Implementation of High-Performance Multiplier

I proposed a method of using full-custom design 32 × 32 multiplier to enhance performance, reduce the power consumption and the area of layout. I use improved Wallace tree structure for partial product compression, truncated beyond the 64 part of the plot and the look-ahead logarithmic adder using Radix-4 Kogge-Stone tree algorithm raise the multiplier performance. In the design of Booth2 encoding circuit and compression circuit, I use a transmission gate logic design with higher speed and smaller area. I also use Euler path method and heuristic Euler path method to reduce the layout area. The design use SMIC 0.18μm 1P4M CMOS process, with a layout area of 0.1684mm2. In a large number of test patterns, simulation results show that the computation time of a 32 × 32 multiplication is less than 3.107ns.

Some natural decision problems in automatic graphs

For automatic and recursive graphs, we investigate the following problems:(A) existence of a Hamiltonian path and existence of an infinite path in a tree(B) existence of an Euler path, bounding the number of ends, and bounding the number of infinite branches in a tree(C) existence of an infinite clique and an infinite version of set coverThe complexity of these problems is determined for automatic graphs and. supplementing results from the literature, for recursive graphs. Our results show that these problems(A) are equally complex for automatic and for recursive graphs (-complete).(B) are moderately less complex for automatic than for recursive graphs (complete for different levels of the arithmetic hierarchy),(C) are much simpler for automatic than for recursive graphs (decidable and -complete, resp.).